View source for Self-injective ring

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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.

====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>



====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" />;

<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>

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" />.

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.

====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>