A characterization of algebras of invariant-coinvariant module type

Author:
Ryohei Makino

Journal:
Proc. Amer. Math. Soc. **92** (1984), 10-12

MSC:
Primary 16A48; Secondary 16A64

DOI:
https://doi.org/10.1090/S0002-9939-1984-0749880-1

MathSciNet review:
749880

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: K. R. Fuller has characterized rings of left invariant module type. An algebra is said to be of invariant-coinvariant module type if each of its indecomposable modules is quasi-injective or quasi-projective. In this note we shall give a characterization of algebras of invariant-coinvariant module type, which distinguishes this class from that of algebras of local-colocal type. It seems of interest that the distributivity of second radicals of primitive ideals appears in our characterization.

**[1]**V. P. Camillo,*Distributive modules*, J. Algebra**36**(1975), 16-25. MR**0573061 (58:28076)****[2]**S. E. Dickson and K. R. Fuller,*Algebras for which every indecomposable right module is invariant in its injective envelope*, Pacific J. Math.**31**(1969), 655-658. MR**0252433 (40:5653)****[3]**K. R. Fuller,*Rings of left invariant module type*, Comm. Algebra**6**(1978), 153-167. MR**0472908 (57:12593)****[4]**-,*Biserial rings*, Lecture Notes in Math., vol. 734, Springer-Verlag, Berlin and New York, 1979, pp. 64-90. MR**548124 (82m:16012)****[5]**R. E. Johnson and E. T. Wong,*Quasi-injective modules and irreducible rings*, J. London Math. Soc.**36**(1961), 260-268. MR**0131445 (24:A1295)****[6]**H. Tachikawa,*On rings for which every indecomposable right module has a unique maximal submodule*, Math. Z.**71**(1959), 200-222. MR**0107662 (21:6385)****[7]**-,*On algebras of which every indecomposable representation has an irreducible one as the top or the bottom Loewy constituent*, Math. Z.**75**(1961), 215-227. MR**0124356 (23:A1670)****[8]**L. E. T. Wu and J. P. Jans,*On quasi projectives*, Illinois J. Math.**11**(1967), 439-448. MR**0220765 (36:3817)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
16A48,
16A64

Retrieve articles in all journals with MSC: 16A48, 16A64

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1984-0749880-1

Keywords:
Algebra of local-colocal type,
distributive module,
quasi-injective module,
quasi-projective module

Article copyright:
© Copyright 1984
American Mathematical Society