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In algebra, '''Auslander–Reiten theory''' studies the representation theory of [[Artinian ring]]s using techniques such as '''Auslander–Reiten sequences''' (also called '''almost split sequences''') and '''Auslander–Reiten quivers'''. Auslander–Reiten theory was introduced by {{harvs|txt|first=Maurice |last=Auslander|author1-link=Maurice Auslander|first2=Idun |last2=Reiten|author2-link=Idun Reiten|year=1975}} and developed by them in several subsequent papers.
 
For survey articles on Auslander–Reiten theory see {{harvtxt|Auslander|1982}}, {{harvtxt|Gabriel|1980}}, {{harvtxt|Reiten|1982}}, and the book {{harvtxt|Auslander|Reiten|Smalø|1997}}. Many of the original papers on Auslander–Reiten theory are reprinted in  {{harvs|txt|last=Auslander|year1=1999a|year2=1999b}}.
 
==Almost-split sequences==
 
Suppose that ''R'' is an Artinian algebra. A sequence
:0&rarr; ''A'' &rarr; ''B'' &rarr; ''C'' &rarr; 0
of finitely generated left modules over ''R'' is called an '''almost-split sequence''' (or '''Auslander–Reiten sequence''') if it has the following properties:
*The sequence is not split
*''C'' is indecomposable and any homomorphism from an indecomposable module to ''C'' that is not an isomorphism factors through ''B''.
*''A'' is indecomposable and any homomorphism from ''A'' to an indecomposable module that is not an isomorphism factors through ''B''.
 
For any finitely generated left module ''C'' that is indecomposable but not projective there is an almost-split sequence as above, which is unique up to isomorphism. Similarly for any finitely generated left module ''A'' that is indecomposable but not injective there is an almost-split sequence as above, which is unique up to isomorphism.
 
The module ''A'' in the almost split sequence is isomorphic to D Tr ''C'', the [[Artin algebra|dual]] of the [[Artin algebra|transpose]] of ''C''.
 
===Example===
 
Suppose that ''R'' is the ring ''k''[''x'']/(''x''<sup>''n''</sup>) for a field ''k'' and an integer ''n''≥1. The indecomposable modules are isomorphic to one of ''k''[''x'']/(''x''<sup>''m''</sup>) for 1≤ ''m'' ≤ ''n'', and the only projective one has ''m''=''n''. The almost split sequences are isomorphic to
:<math> 0 \rightarrow k[x]/(x^m) \rightarrow k[x]/(x^{m+1}) \oplus k[x]/(x^{m-1}) \rightarrow k[x]/(x^{m}) \rightarrow 0</math>
for 1 ≤ ''m'' < ''n''. The first morphism takes ''a'' to (''xa'', ''a'') and the second takes (''b'',''c'') to&nbsp;''b''&nbsp;&minus;&nbsp;''xc''.
 
==Auslander-Reiten quiver==
 
The '''Auslander-Reiten quiver''' of an Artin algebra has a vertex for each indecomposable module and an arrow between vertices if there is an irreducible morphism between the corresponding modules. It has a map τ = ''D Tr'' called the '''translation''' from the non-projective vertices to the non-injective vertices, where ''D'' is the dual and ''Tr'' the transpose.
 
==References==
 
*{{Citation | last1=Auslander | first1=Maurice | title=Representations of algebras (Puebla, 1980) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | doi=10.1007/BFb0094058 | mr=672116 | year=1982 | volume=944 | chapter=A functorial approach to representation theory | pages=105–179}}
*{{Citation | last1=Auslander | first1=Maurice | title=Proceedings of the International Congress of Mathematicians, Vol. 1 (Berkeley, Calif., 1986) | url=http://www.mathunion.org/ICM/ICM1986.1/ | publisher=Amer. Math. Soc. | location=Providence, R.I. | mr=934232 | year=1987 | chapter=The what, where, and why of almost split sequences | pages=338–345}}
*{{Citation | last1=Auslander | first1=Maurice | last2=Reiten | first2=Idun | last3=Smalø | first3=Sverre O. | title=Representation theory of Artin algebras | origyear=1995 | url=http://books.google.com/books?isbn=0521599237 | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-59923-8 | mr=1314422 | year=1997 | volume=36}}
*{{Citation | last1=Auslander | first1=Maurice | editor1-last=Reiten | editor1-first=Idun | editor2-last=Smalø | editor2-first=Sverre O. | editor3-last=Solberg | editor3-first=Øyvind | title=Selected works of Maurice Auslander. Part 1 | url=http://books.google.com/books?isbn=0821809989 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-0998-3 | mr=1674397 | year=1999a}}
*{{Citation | last1=Auslander | first1=Maurice | editor1-last=Reiten | editor1-first=Idun | editor2-last=Smalø | editor2-first=Sverre O. | editor3-last=Solberg | editor3-first=Øyvind | title=Selected works of Maurice Auslander. Part 2 | url=http://books.google.com/books?isbn=0821810006 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | isbn=978-0-8218-1000-2 | mr=1674401 | year=1999b}}
*{{Citation | last1=Auslander | first1=Maurice | last2=Reiten | first2=Idun | title=Representation theory of Artin algebras. III. Almost split sequences | doi=10.1080/00927877508822046  | mr=0379599 | year=1975 | journal=Communications in Algebra | issn=0092-7872 | volume=3 | issue=3 | pages=239–294}}
*{{Citation | last1=Gabriel | first1=Peter | editor1-last=Dlab | editor1-first=Vlastimil | editor2-last=Gabriel | editor2-first=Peter | title=Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | doi=10.1007/BFb0089778 | mr=607140 | year=1980 | volume=831 | chapter=Auslander-Reiten sequences and representation-finite algebras | pages=1–71}}
*{{eom|id=A/a130220|title=Almost-split sequence|first=M.|last= Hazewinkel}}
*{{Citation | last1=Reiten | first1=Idun | title=Representations of algebras (Puebla, 1980) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | doi=10.1007/BFb0094057 | mr=672115 | year=1982 | volume=944 | chapter=The use of almost split sequences in the representation theory of Artin algebras | pages=29–104}}
 
==External links==
*{{citation|url=http://www.math.jussieu.fr/~keller/ictp2006/lecturenotes/angeleri.pdf |title=An Introduction to Auslander-Reiten Theory  |first=Lidia Angeleri|last= Hugel|year=2006}}
 
{{DEFAULTSORT:Auslander-Reiten theory}}
[[Category:Representation theory]]

Revision as of 14:17, 12 April 2013

In algebra, Auslander–Reiten theory studies the representation theory of Artinian rings using techniques such as Auslander–Reiten sequences (also called almost split sequences) and Auslander–Reiten quivers. Auslander–Reiten theory was introduced by Template:Harvs and developed by them in several subsequent papers.

For survey articles on Auslander–Reiten theory see Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, and the book Template:Harvtxt. Many of the original papers on Auslander–Reiten theory are reprinted in Template:Harvs.

Almost-split sequences

Suppose that R is an Artinian algebra. A sequence

0→ ABC → 0

of finitely generated left modules over R is called an almost-split sequence (or Auslander–Reiten sequence) if it has the following properties:

  • The sequence is not split
  • C is indecomposable and any homomorphism from an indecomposable module to C that is not an isomorphism factors through B.
  • A is indecomposable and any homomorphism from A to an indecomposable module that is not an isomorphism factors through B.

For any finitely generated left module C that is indecomposable but not projective there is an almost-split sequence as above, which is unique up to isomorphism. Similarly for any finitely generated left module A that is indecomposable but not injective there is an almost-split sequence as above, which is unique up to isomorphism.

The module A in the almost split sequence is isomorphic to D Tr C, the dual of the transpose of C.

Example

Suppose that R is the ring k[x]/(xn) for a field k and an integer n≥1. The indecomposable modules are isomorphic to one of k[x]/(xm) for 1≤ mn, and the only projective one has m=n. The almost split sequences are isomorphic to

for 1 ≤ m < n. The first morphism takes a to (xa, a) and the second takes (b,c) to b − xc.

Auslander-Reiten quiver

The Auslander-Reiten quiver of an Artin algebra has a vertex for each indecomposable module and an arrow between vertices if there is an irreducible morphism between the corresponding modules. It has a map τ = D Tr called the translation from the non-projective vertices to the non-injective vertices, where D is the dual and Tr the transpose.

References

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

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    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010