# Balanced ring

From Encyclopedia of Mathematics

2010 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

This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article