Difference between revisions of "Balanced ring"
From Encyclopedia of Mathematics
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| − | A ring over which all left (right) modules are balanced. A ring is balanced on the left if and only if all its quotient rings are QF-$1$-rings, that is, if all the exact left modules over it are balanced. In particular, a ring is balanced if all its quotient rings are quasi-Frobenius. Every balanced ring can be split into a direct sum of a uniserial ring and rings of matrices over local rings of a special type. Every balanced ring is semi-perfect. A Noetherian balanced ring is an Artinian ring. | + | A ring over which all left (right) modules are [[Balanced module|balanced]]. A ring is balanced on the left if and only if all its quotient rings are QF-$1$-rings, that is, if all the exact left modules over it are balanced. In particular, a ring is balanced if all its quotient rings are [[Quasi-Frobenius ring|quasi-Frobenius]]. Every balanced ring can be split into a direct sum of a [[uniserial ring]] and rings of matrices over local rings of a special type. Every balanced ring is [[Semi-perfect ring|semi-perfect]]. A [[Noetherian ring|Noetherian]] balanced ring is an [[Artinian ring]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''19''' (1981) pp. 31–134</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> C. Faith, "Algebra" , '''1–2''' , Springer (1973–1976)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''19''' (1981) pp. 31–134</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> C. Faith, "Algebra" , '''1–2''' , Springer (1973–1976)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 18:25, 11 December 2014
2020 Mathematics Subject Classification: Primary: 16D [MSN][ZBL]
on the left (right)
A ring over which all left (right) modules are balanced. A ring is balanced on the left if and only if all its quotient rings are QF-$1$-rings, that is, if all the exact left modules over it are balanced. In particular, a ring is balanced if all its quotient rings are quasi-Frobenius. Every balanced ring can be split into a direct sum of a uniserial ring and rings of matrices over local rings of a special type. Every balanced ring is semi-perfect. A Noetherian balanced ring is an Artinian ring.
References
| [1] | Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 19 (1981) pp. 31–134 |
| [2] | C. Faith, "Algebra" , 1–2 , Springer (1973–1976) |
How to Cite This Entry:
Balanced ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Balanced_ring&oldid=35544
Balanced ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Balanced_ring&oldid=35544
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article