Modules which are invariant under nilpotents of their envelopes and covers

A module is called nilpotent-invariant if it is invariant under any nilpotent endomorphism of its injective envelope [M. T. Koşan and T. C. Quynh, Nilpotent-invaraint modules and rings, Comm. Algebra 45 (2017) 2775–2782]. In this paper, we continue the study of nilpotent-invariant modules and analyze their relationship to (quasi-)injective modules. It is proved that a right module [Formula: see text] over a semiprimary ring is nilpotent-invariant iff all nilpotent endomorphisms of submodules of [Formula: see text] extend to nilpotent endomorphisms of [Formula: see text]. It is also shown that a right module [Formula: see text] over a prime right Goldie ring with [Formula: see text] is nilpotent-invariant iff it is injective. We also study nilpotent-coinvariant modules that are the dual notation of nilpotent-invariant modules. It is proved that if [Formula: see text] is a finitely generated nilpotent-coinvariant right module with [Formula: see text] square-full, then [Formula: see text] is quasi-projective. Some characterizations and structures of nilpotent-coinvariant modules are considered.

On semirings all of whose semimodules have injective envelopes

It was shown in [Y. Katsov, Tensor products and injective envelopes of semimodules over additively regular semirings, Algebra Colloq. 4 (1997) 121–131.] that each semimodule over an additively regular semiring has an injective envelope. We show the converse statement is true as well; moreover, the result holds even if each finitely generated semimodule has an injective envelope.

Neat-phantom and clean-cophantom morphisms

Let [Formula: see text] be the class of all left [Formula: see text]-modules [Formula: see text] which has a projective resolution by finitely generated projectives. An exact sequence [Formula: see text] of right [Formula: see text]-modules is called neat if the sequence [Formula: see text] is exact for any [Formula: see text]. An exact sequence [Formula: see text] of left [Formula: see text]-modules is called clean if the sequence [Formula: see text] is exact for any [Formula: see text]. We prove that every [Formula: see text]-module has a clean-projective precover and a neat-injective envelope. A morphism [Formula: see text] of right [Formula: see text]-modules is called a neat-phantom morphism if [Formula: see text] for any [Formula: see text]. A morphism [Formula: see text] of left [Formula: see text]-modules is said to be a clean-cophantom morphism if [Formula: see text] for any [Formula: see text]. We establish the relationship between neat-phantom (respectively, clean-cophantom) morphisms and neat (respectively, clean) exact sequences. Also, we prove that every [Formula: see text]-module has a neat-phantom cover with kernel neat-injective and a clean-cophantom preenvelope with cokernel clean-projective.

Injective modules over the Jacobson algebra

For a field K, let $\mathcal {R}$ denote the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle $ . We give an explicit construction of the injective envelope of each of the (infinitely many) simple left $\mathcal {R}$ -modules. Consequently, we obtain an explicit description of a minimal injective cogenerator for $\mathcal {R}$ . Our approach involves realizing $\mathcal {R}$ up to isomorphism as the Leavitt path K-algebra of an appropriate graph $\mathcal {T}$ , which thereby allows us to utilize important machinery developed for that class of algebras.

Injective envelopes of transition systems and Ferrers languages

We consider reflexive and involutive transition systems over an ordered alphabet A equipped with an involution. We give a description of the injective envelope of any two-element set in terms of Galois lattice, from which we derive a test of its finiteness. Our description leads to the notion of Ferrers language.

The Non-selfadjoint Approach to the Hao–Ng Isomorphism

In an earlier work, the authors proposed a non-selfadjoint approach to the Hao–Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of $\mathrm{C}^*$-correspondences, each one of these conjectures is equivalent to the Hao–Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C$^*$-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of $\mathrm{C}^*$-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana’s injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao–Ng isomorphism for the reduced crossed product and all hyperrigid $\mathrm{C}^*$-correspondences. A culmination of these results is the resolution of the Hao–Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg, and Spielberg.

T-idempotent invariant modules

We introduce and investigate [Formula: see text]-idempotent invariant modules. We call an endomorphism [Formula: see text] of [Formula: see text], a [Formula: see text]-idempotent endomorphism if [Formula: see text] defined by [Formula: see text] is an idempotent and we call a module [Formula: see text] is [Formula: see text]-idempotent invariant, if it is invariant under [Formula: see text]-idempotents of its injective envelope. We prove a module [Formula: see text] is [Formula: see text]-idempotent invariant if and only if [Formula: see text], [Formula: see text] is quasi-injective, [Formula: see text] is quasi-continuous and [Formula: see text] is [Formula: see text]-injective. The class of rings [Formula: see text] for which every (finitely generated, cyclic, free) [Formula: see text]-module is [Formula: see text]-idempotent invariant is characterized. Moreover, it is proved that if [Formula: see text] is right q.f.d., then every [Formula: see text]-idempotent invariant [Formula: see text]-module is quasi-injective exactly when every nonsingular uniform [Formula: see text]-module is quasi-injective.

2-Nilpotent-invariant modules

A module which is invariant under automorphisms of its injective envelope is called an automorphism-invariant module. The class of automorphism-invariant modules was introduced and investigated by Lee and Zhou in 2013. In this paper, we study the class of modules which are invariant under all nilpotent endomorphisms of their injective envelopes of index two, such as modules are called 2-nilpotent-invariant. Many basic properties are obtained. For instance, it is proved that a nonsingular module [Formula: see text] is a weak duo 2-nilpotent-invariant module if and only if [Formula: see text] is a strongly regular ring. For the ring [Formula: see text] satisfying every cyclic right [Formula: see text]-module is 2-nilpotent-invariant, we prove that [Formula: see text], where [Formula: see text] are rings which satisfy [Formula: see text] is a semi-simple Artinian ring and [Formula: see text] is square-free as a right [Formula: see text]-module and all idempotents of [Formula: see text] is central.

On injective envelopes of semimodules over semirings

As is well-known, every module over a ring has an injective envelope; however, that fact does not hold, in general, for semimodules over semirings. We study the problem of existence of injective envelopes with respect to several types of semimodules, in particular, we consider simple semimodules, modules, additively idempotent semimodules, and additively regular semimodules. We completely describe semirings whose all semimodules of these types have injective envelopes.