A ring R satisfies the dual of the isomorphism theorem if R/Ra ∼ = l(a) for all elements a of R, where l(a) denotes the left annihilator. We call these rings left morphic. Examples include all unit regular rings and certain left uniserial local rings. We show that every left morphic ring is right principally injective, and use this to characterize the left perfect, right and left morphic rings. 2004 Elsevier Inc. All rights reserved.It is a well-known theorem of Erlich [5] that a map α in the endomorphism ring of a module M is unit regular if and only if it is regular and M/im(α) ∼ = ker(α). Our focus is on the case M = R R, so if α = · a : R R → R R is right multiplication by the element a ∈ R, the condition becomes R/Ra ∼ = l(a) where l(a) denotes the left annihilator. We say that the ring R is left morphic if every element satisfies this condition. We begin (Theorem 9) by characterizing the left morphic, local rings with nilpotent radical (and call these rings left 'special'); in particular we show that these rings are all left artinian. We show (Theorem 29) that a semiperfect left morphic ring is a finite product of matrix rings over local left morphic rings, we use this result to characterize (in Theorem 35) the left perfect, left and right morphic rings as the finite products of matrix rings over left and right 'special' rings.Along the way, we show (Theorem 24) that every left morphic ring is right principally injective [11]. With this we see that the left morphic rings R with ACC on right annihilators are left artinian, and have the property that eRe is left 'special' for every local idempotent e
Abstract.A ring R satisfies the dual of the isomorphism theorem if R/Ra ∼ = l(a) for every element a ∈ R. We call these rings left morphic, and say that R is left P-morphic if, in addition, every left ideal is principal. In this paper we characterize the left and right P-morphic rings and show that they form a Morita invariant class. We also characterize the semiperfect left P-morphic rings as the finite direct products of matrix rings over left uniserial rings of finite composition length. J. Clark has an example of a commutative, uniserial ring with exactly one non-principal ideal. We show that Clark's example is (left) morphic and obtain a non-commutative analogue.2000 Mathematics Subject Classification. Primary 16E50, secondary 16U99, 16S70. Introduction.If R is a ring, the isomorphism theorem asserts that R/l(a) ∼ = Ra for every element a ∈ R (where l(a) denotes the left annihilator). Dually, R is called left morphic if it satisfies the following equivalent conditions [6, Lemma 1].• R/Ra ∼ = l(a) for every a ∈ R;• For every a ∈ R there exists b ∈ R such that Ra = l(b) and l(a) = Rb. Every unit regular ring is left and right morphic and ޚ 4 is an example of a nonregular morphic ring. We call a ring R left special if it satisfies the following equivalent conditions [6, Theorem 9].• R is left uniserial of finite length;• R is left morphic, local and the Jacobson radical J is nilpotent;• R is local and J = Rc where c ∈ R is nilpotent. In this case J n = Rc n for each n ≥ 0. The ring ޚ p n of integers modulo p n is a commutative special ring for every prime p, and other examples appear below.The left morphic rings were studied in [6] where, among other things, the left artinian, left and right morphic rings were characterized as the finite products of matrix rings over left and right special rings. Call a ring R left principally morphic (left P-morphic) if it is left morphic and every left ideal is principal. The local, left P-morphic rings are just the left special rings, but matrix rings over left P-morphic rings need not be left morphic. Nonetheless, we prove the following results.
An R-module M is called strongly Hopfian (respectively strongly co-Hopfian) if for every endomorphism f of M the chain Ker f ⊆ Ker f 2 ⊆ · · · (respectively Im f ⊇ Im f 2 ⊇ · · ·) stabilizes. The class of strongly Hopfian (respectively co-Hopfian) modules lies properly between the class of Noetherian (respectively Artinian) and the class of Hopfian (respectively co-Hopfian) modules. For a quasi-projective (respectively quasi-injective) module, M, if M is strongly co-Hopfian (respectively strongly Hopfian) then M is strongly Hopfian (respectively strongly co-Hopfian). As a consequence we obtain a version of Hopkins-Levitzki theorem for strongly co-Hopfian rings. Namely, a strongly co-Hopfian ring is strongly Hopfian. Also we prove that for a commutative ring A, the polynomial ring A[X] is strongly Hopfian if and only if A is strongly Hopfian.
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