On the idempotence and stability of kernel functors

Abstract

Introduction.A kernel functor (equivalently, a left exact torsion preradical) is a left exact subfunctor of the identity on the category i?-mod of left R -modules over a ring R with identity. A kernel functor is said to be idempotent if, in addition, <r satisfies er(A//o-(M)) = 0 for every M e 7?-mod. To every kernel functor a there corresponds a unique topologizing filter %" = {! \a(R/I) = R/I] of left ideals and a unique class by showing that the questions about the idempotence center around the stability of the Gabriel kernel functor. In Theorem 3 we show that when every kernel functor for i?-mod is idempotent, R = R l ®R 2 (ring direct sum), where R^ is a ring with Gabriel dimension and R 2 has no nonzero ideals with Gabriel dimension. This separates the idempotence question into the study of rings with Gabriel dimension and rings without Gabriel dimension. In Theorem 4, we show that if R has Gabriel dimension and every kernel functor for R-mod is idempotent, then R is left noetherian. In Theorem 7 we give several characterizations of when a ring with Gabriel dimension satisfies the condition: every kernel functor for i?-mod is idempotent. One of the equivalent conditions is that every kernel functor for /?-mod is stable. This property has independent interest in view of the paper [11] by Papp and subsequent papers by Papp and others (e.g., see [3] and [5, Chapter 50]) on the question: when is every idempotent kernel functor stable? Finally, in the spirit of [15, Theorem 2.6], we give two decomposition theorems for rings satisfying the conditions:(1) every kernel functor for 7?-mod is stable, or (2) every kernel functor for fl-mod is idempotent and the Gabriel kernel functor fi is stable.If every kernel functor for i?-mod is idempotent, it is well known ([6] or [14]) and easy to prove that R is a left nonsingular semiprime ring such that I 2 = I for every two-sided ideal / of R.Let G-dim M denote the Gabriel dimension of a left R-module M (when it exists).

When every torsion preradical is a torsion radical

“…We point out t,hat 7~c r > r V u for all 7,cr E t,orsp R. If C is a mnempty class of right, R-modules we d e~~o t , e by c~c t,he smallest torsion preradical on Mod-R r e h i v e t,o which every member of C is t,orsion. If M E Mod-R then M is uc-t,orsion if and only if M is a homornorphic image of a submodule of a direct sum of modules in C (see [13,Lemma. 1,p38]).…”

Primeness described in the language of torsion preradicals

“…If C is a nonempty class of right R -modules we denote by σ C the smallest torsion preradical on Mod-R relative to which every member of C is torsion. If M ∈ Mod-R then M is σ C -torsion if and only if M is a homomorphic image of a submodule of a direct sum of modules in C (see [11,Lemma 1,p. 38]).…”