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A ring that, as a left module over itself, is injective (cf. [[Injective module|Injective module]]). A right self-injective ring is defined in a symmetric way. The classical semi-simple rings and all rings of residues of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s0839001.png" /> are self-injective rings. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s0839002.png" /> is a self-injective ring with [[Jacobson radical|Jacobson radical]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s0839003.png" />, then the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s0839004.png" /> is a [[Regular ring (in the sense of von Neumann)|regular ring (in the sense of von Neumann)]]. A regular self-injective ring is continuous. Every countable self-injective ring is quasi-Frobenius (cf. [[Quasi-Frobenius ring|Quasi-Frobenius ring]]). A left self-injective ring is not necessarily right self-injective. The ring of matrices over a self-injective ring and the complete ring of linear transformations of a vector space over a field are self-injective. The rings of endomorphisms of all free left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s0839005.png" />-modules are self-injective rings if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s0839006.png" /> is quasi-Frobenius. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s0839007.png" /> is the cogenerator of the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s0839008.png" />-modules, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s0839009.png" /> is a self-injective ring. If the singular ideal of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390010.png" /> is zero, then its [[injective hull]] can be made into a self-injective ring in a natural way. A group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390011.png" /> is left self-injective if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390012.png" /> is a self-injective ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390013.png" /> is a finite group. The direct product of self-injective rings is self-injective. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390014.png" /> is isomorphic to the direct product of complete rings of linear transformations over fields if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390015.png" /> is a left self-injective ring without nilpotent ideals for which every non-zero left ideal contains a minimal left ideal.
+
A ring that, as a left module over itself, is injective (cf. [[Injective module|Injective module]]). A right self-injective ring is defined in a symmetric way. The classical semi-simple rings and all rings of residues of integers $  \mathbf Z /( n) $
 +
are self-injective rings. If $  R $
 +
is a self-injective ring with [[Jacobson radical|Jacobson radical]] $  J $,  
 +
then the quotient ring $  R/J $
 +
is a [[Regular ring (in the sense of von Neumann)|regular ring (in the sense of von Neumann)]]. A regular self-injective ring is continuous. Every countable self-injective ring is quasi-Frobenius (cf. [[Quasi-Frobenius ring|Quasi-Frobenius ring]]). A left self-injective ring is not necessarily right self-injective. The ring of matrices over a self-injective ring and the complete ring of linear transformations of a vector space over a field are self-injective. The rings of endomorphisms of all free left $  R $-
 +
modules are self-injective rings if and only if $  R $
 +
is quasi-Frobenius. If $  M $
 +
is the cogenerator of the category of left $  R $-
 +
modules, then $  \mathop{\rm End} _ {R}  M $
 +
is a self-injective ring. If the singular ideal of a ring $  R $
 +
is zero, then its [[injective hull]] can be made into a self-injective ring in a natural way. A group ring $  RG $
 +
is left self-injective if and only if $  R $
 +
is a self-injective ring and $  G $
 +
is a finite group. The direct product of self-injective rings is self-injective. A ring $  R $
 +
is isomorphic to the direct product of complete rings of linear transformations over fields if and only if $  R $
 +
is a left self-injective ring without nilpotent ideals for which every non-zero left ideal contains a minimal left ideal.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Skornyaka,  A.V. Mikhalev,  "Modules"  ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''14'''  (1976)  pp. 57–190  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Faith,  "Algebra" , '''1–2''' , Springer  (1973–1976)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Lawrence,  "A countable self-injective ring is quasi-Frobenius"  ''Proc. Amer. Math. Soc.'' , '''65''' :  2  (1977)  pp. 217–220</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Skornyaka,  A.V. Mikhalev,  "Modules"  ''Itogi Nauk. i Tekhn. Algebra. Topol. Geom.'' , '''14'''  (1976)  pp. 57–190  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Faith,  "Algebra" , '''1–2''' , Springer  (1973–1976)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Lawrence,  "A countable self-injective ring is quasi-Frobenius"  ''Proc. Amer. Math. Soc.'' , '''65''' :  2  (1977)  pp. 217–220</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
An essential right ideal of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390016.png" /> is an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390018.png" /> for all non-zero right ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390020.png" />. In a right Ore domain (cf. below) every non-zero right ideal is essential. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390021.png" /> be the set of essential right ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390022.png" />;
+
An essential right ideal of a ring $  R $
 +
is an ideal $  E $
 +
such that $  E \cap I \neq 0 $
 +
for all non-zero right ideals $  I $
 +
of $  R $.  
 +
In a right Ore domain (cf. below) every non-zero right ideal is essential. Let $  {\mathcal E} ( R) $
 +
be the set of essential right ideals of $  R $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390023.png" /></td> </tr></table>
+
$$
 +
\zeta ( R)  = \{ {a \in R } : {a E = 0 \textrm{ for  some  }
 +
E \in {\mathcal E} ( R) } \}
 +
$$
  
is an ideal, called the right singular ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390024.png" />.
+
is an ideal, called the right singular ideal of $  R $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390025.png" /> be the multiplicatively closed subset of regular elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390026.png" /> (i.e. non-zero-divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390027.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390028.png" /> satisfies the right Ore condition (cf. [[Associative rings and algebras|Associative rings and algebras]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083900/s08390029.png" /> is called a right Ore ring. A right Ore domain is an [[Integral domain|integral domain]] that is a right Ore ring.
+
Let $  S $
 +
be the multiplicatively closed subset of regular elements of $  R $(
 +
i.e. non-zero-divisors of $  R $).  
 +
If $  S $
 +
satisfies the right Ore condition (cf. [[Associative rings and algebras|Associative rings and algebras]]), $  R $
 +
is called a right Ore ring. A right Ore domain is an [[Integral domain|integral domain]] that is a right Ore ring.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C. McConnell,  J.C. Robson,  "Noncommutative Noetherian rings" , Wiley  (1987)  pp. Part I, Chapt. 2</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C. McConnell,  J.C. Robson,  "Noncommutative Noetherian rings" , Wiley  (1987)  pp. Part I, Chapt. 2</TD></TR></table>

Latest revision as of 08:13, 6 June 2020


left

A ring that, as a left module over itself, is injective (cf. Injective module). A right self-injective ring is defined in a symmetric way. The classical semi-simple rings and all rings of residues of integers $ \mathbf Z /( n) $ are self-injective rings. If $ R $ is a self-injective ring with Jacobson radical $ J $, then the quotient ring $ R/J $ is a regular ring (in the sense of von Neumann). A regular self-injective ring is continuous. Every countable self-injective ring is quasi-Frobenius (cf. Quasi-Frobenius ring). A left self-injective ring is not necessarily right self-injective. The ring of matrices over a self-injective ring and the complete ring of linear transformations of a vector space over a field are self-injective. The rings of endomorphisms of all free left $ R $- modules are self-injective rings if and only if $ R $ is quasi-Frobenius. If $ M $ is the cogenerator of the category of left $ R $- modules, then $ \mathop{\rm End} _ {R} M $ is a self-injective ring. If the singular ideal of a ring $ R $ is zero, then its injective hull can be made into a self-injective ring in a natural way. A group ring $ RG $ is left self-injective if and only if $ R $ is a self-injective ring and $ G $ is a finite group. The direct product of self-injective rings is self-injective. A ring $ R $ is isomorphic to the direct product of complete rings of linear transformations over fields if and only if $ R $ is a left self-injective ring without nilpotent ideals for which every non-zero left ideal contains a minimal left ideal.

References

[1] L.A. Skornyaka, A.V. Mikhalev, "Modules" Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 14 (1976) pp. 57–190 (In Russian)
[2] C. Faith, "Algebra" , 1–2 , Springer (1973–1976)
[3] J. Lawrence, "A countable self-injective ring is quasi-Frobenius" Proc. Amer. Math. Soc. , 65 : 2 (1977) pp. 217–220

Comments

An essential right ideal of a ring $ R $ is an ideal $ E $ such that $ E \cap I \neq 0 $ for all non-zero right ideals $ I $ of $ R $. In a right Ore domain (cf. below) every non-zero right ideal is essential. Let $ {\mathcal E} ( R) $ be the set of essential right ideals of $ R $;

$$ \zeta ( R) = \{ {a \in R } : {a E = 0 \textrm{ for some } E \in {\mathcal E} ( R) } \} $$

is an ideal, called the right singular ideal of $ R $.

Let $ S $ be the multiplicatively closed subset of regular elements of $ R $( i.e. non-zero-divisors of $ R $). If $ S $ satisfies the right Ore condition (cf. Associative rings and algebras), $ R $ is called a right Ore ring. A right Ore domain is an integral domain that is a right Ore ring.

References

[a1] J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) pp. Part I, Chapt. 2
How to Cite This Entry:
Self-injective ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Self-injective_ring&oldid=48650
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article