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Difference between revisions of "Balanced ring"

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''on the left (right)''
 
''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-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015090/b0150902.png" />-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.
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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.
  
 
====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>

Revision as of 14:21, 12 April 2014

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=31637
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article