Difference between revisions of "Self-injective ring"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
(One intermediate revision by one other user not shown) | |||
Line 1: | Line 1: | ||
+ | <!-- | ||
+ | s0839001.png | ||
+ | $#A+1 = 29 n = 0 | ||
+ | $#C+1 = 29 : ~/encyclopedia/old_files/data/S083/S.0803900 Self\AAhinjective ring, | ||
+ | Automatically converted into TeX, above some diagnostics. | ||
+ | Please remove this comment and the {{TEX|auto}} line below, | ||
+ | if TeX found to be correct. | ||
+ | --> | ||
+ | |||
+ | {{TEX|auto}} | ||
+ | {{TEX|done}} | ||
+ | |||
''left'' | ''left'' | ||
− | 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 | + | 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 | + | 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 | + | is an ideal, called the right singular ideal of $ R $. |
− | Let | + | 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 |
Self-injective ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Self-injective_ring&oldid=17113