Rings with Indecomposable Right Modules Uniform

Abstract

A ring R is said to satisfy condition (**) if it is right artinian and every indecomposable right R-module of finite composition length is uniform. Such rings were first studied by Tachikawa [7]. The approach here is quite different from that of Tachikawa. To start with a theorem on lifting of an isomorphism between simple homomorphic images of two uniform modules over a ring satisfying (**) is proved. Let M be a uniform module of finite composition length with S = soc(M), D = End(S) and D′ be the division subring of D consisting of those σ ∈D which can be extended to some endomorphism of M. Then the pair (D, D′) is called the drpa of M. The following is proved. If a ring R with Jacobson radical J satisfies (**), then it satisfies the following conditions: (1) R is a right artinian, right serial ring; (2) for any three indecomposable idempotents e, f, g ∈ R with eJ, fJ, gJ nonzero the following hold: (i) if (D, D′) is the drpa of , then the left dimension of D over D′ is less than or equal to 2; (ii) if e, f are non-isomorphic and , then eJ ² = 0 or fJ ² = 0; (iii) if e, f are non-isomorphic and , then g is isomorphic to e or f; (iv) if is not quasi-injective, then eJ ² = 0 and , whenever e is not isomorphic to f. If R satisfies the above conditions and for any indecomposable idempotent e ∈ R with eJ ≠ 0, the drpa (D, D′) of satisfies [D: D′] _r  ≤ 2, then R satisfies (**).

Key Words:

• Injective modules
• Quasi-projective modules
• Right artinian rigs
• Right serial rings
• Socle series

2010 Mathematics Subject Classification:

• Primary 16G10
• Secondary 16P20

ACKNOWLEDGMENT

This work has been supported by King Saud University, Riyadh, vide the research grant DSFP/MATH1. The author is extremely thankful to the referee for various suggestions.

Notes

Communicated by E. Kirkman.