Finitedimensional right duo algebras are duo
Author:
R. C. Courter
Journal:
Proc. Amer. Math. Soc. 84 (1982), 157161
MSC:
Primary 16A15; Secondary 16A46
MathSciNet review:
637159
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Available examples of right (but not left) duo rings include rings without unity element which are two dimensional algebras over finite prime fields. We prove that a right duo ring with unity element which is a finite dimensional algebra over an arbitrary field is a duo ring. This result is obtained as a corollary of a theorem on right duo, right artinian rings with unity: left duoness is equivalent to each right ideal of having equal right and left composition lengths, which is equivalent to the same property on alone. Another result concerns algebras over a field which are semiprimary right duo rings: such an algebra is left duo provided (1) the algebra is finite dimensional modulo its radical and (2) the square of the radical is zero. These two provisions are shown to be essential by examples which are local algebras, duo on one side only.
 [1]
V.
R. Chandran, On duoringsVI, Pure Appl. Math. Sci.
3 (1976), no. 12, 88–92. MR 0407084
(53 #10867)
 [2]
Jean
Dieudonné, Remarks on quasiFrobenius rings, Illinois
J. Math. 2 (1958), 346–354. MR 0097427
(20 #3896)
 [3]
Carl
Faith, Algebra. II, SpringerVerlag, BerlinNew York, 1976.
Ring theory; Grundlehren der Mathematischen Wissenschaften, No. 191. MR 0427349
(55 #383)
 [4]
Enzo
R. Gentile, On rings with onesided field of
quotients, Proc. Amer. Math. Soc. 11 (1960), 380–384. MR 0122855
(23 #A187), http://dx.doi.org/10.1090/S00029939196001228555
 [5]
Arun
Vinayak Jategaonkar, Left principal ideal domains, J. Algebra
8 (1968), 148–155. MR 0218387
(36 #1474)
 [1]
 V. R. Chandran, On duo rings. VI, Pure Appl. Math. Sci. 3 (1976), 8992. MR 0407084 (53:10867)
 [2]
 J. Dieudonné, Remarks on quasiFrobenius rings, Illinois J. Math. 2 (1958), 346354. MR 0097427 (20:3896)
 [3]
 C. Faith, Algebra. II. Ring theory, SpringerVerlag, Berlin and New York, 1976. MR 0427349 (55:383)
 [4]
 E. R. Gentile, On rings with onesided field of quotients, Proc. Amer. Math. Soc. 11 (1960), 380384. MR 0122855 (23:A187)
 [5]
 A. V. Jategaonkar, Left principal ideal domains, J. Algebra 8 (1968), 148155. MR 0218387 (36:1474)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
16A15,
16A46
Retrieve articles in all journals
with MSC:
16A15,
16A46
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198206371596
PII:
S 00029939(1982)06371596
Article copyright:
© Copyright 1982
American Mathematical Society
