Ring Endomorphisms with Large Images
نویسندگان
چکیده
The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism σ of a prime one-sided noetherian ringR is injective whenever the image σ(R) contains an essential left ideal L of R. If additionally σ(L) = L, then σ is an automorphism of R. Examples showing that the assumptions imposed on R can not be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated. In this paper we start investigations of endomorphisms of semiprime unital rings R having large images, i.e. endomorphisms σ such that the image σ(R) contains an essential left ideal of R (see Definition 1.7). The motivation for such studies is twofold. Let us recall that a ring (or a module) is called Hopfian (resp. co-Hopfian) if every surjective (resp. injective) endomorphism is injective (resp. surjective). It is well known and easy to prove that noetherian (artinian) modules and rings are Hopfian (co-Hopfian). However, in general, the Hopfian property for modules behaves much better than that of rings. Examples showing a difference in that behaviour can be found in [6], [12], [14], [15]. In case of rings, the set of all endomorphisms has no natural structure of a ring and it seems to be natural to consider some classes of endomorphisms of a ring. Our goal is to investigate how one can weaken Hopfian or co-Hopfian assumptions on a ring endomorphism to conclude that the endomorphism is injective or surjective. We obtained some positive results in this direction (Cf. the second part of the introduction). Surprisingly we could not answer some of the elementary formulated problems. For example we proved that every endomorphism ∗The research was supported by Polish MNiSW grant No. N N201 268435
منابع مشابه
Ring endomorphisms with nil-shifting property
Cohn called a ring $R$ is reversible if whenever $ab = 0,$ then $ba = 0$ for $a,bin R.$ The reversible property is an important role in noncommutative ring theory. Recently, Abdul-Jabbar et al. studied the reversible ring property on nilpotent elements, introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring). In this paper, we extend the CNZ pr...
متن کاملStrongly clean triangular matrix rings with endomorphisms
A ring $R$ is strongly clean provided that every element in $R$ is the sum of an idempotent and a unit that commutate. Let $T_n(R,sigma)$ be the skew triangular matrix ring over a local ring $R$ where $sigma$ is an endomorphism of $R$. We show that $T_2(R,sigma)$ is strongly clean if and only if for any $ain 1+J(R), bin J(R)$, $l_a-r_{sigma(b)}: Rto R$ is surjective. Furt...
متن کاملOplus-supplemented modules with respect to images of a fully invariant submodule
Lifting modules and their various generalizations as some main concepts in module theory have been studied and investigated extensively in recent decades. Some authors tried to present some homological aspects of lifting modules and -supplemented modules. In this work, we shall present a homological approach to -supplemented modules via fully invariant submodules. Lifting modules and H-suppleme...
متن کاملSemirings whose additive endomorphisms are multiplicative
A ring or an idempotent semiring is associative provided that additive endomorphisms are multiplicative.
متن کاملTate Conjecture for Drinfeld Modules in Equal Characteristic
We prove that the ring of endomorphisms of the ℘-divisible group of a Drinfeld module of characteristic ℘ is canonically isomorphic to the ring of endomorphisms of the underlying Drinfeld module, completed in the ℘–adic topology. This completes the proof of the Tate Conjectures in the Drinfeld module setting.
متن کامل