Balanced ring

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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--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.


[1] Itogi Nauk. i Tekhn. Algebra Topol. Geom. , 19 (1981) pp. 31–134
[2] C. Faith, "Algebra" , 1–2 , Springer (1973–1976)
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Balanced ring. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article