• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.289-309
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• 2019

This paper is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commutative property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpotent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.83-91
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• 2023

A ring R is called a UN ring if every non unit of it can be written as product of a unit and a nilpotent element. We obtain results about lifting of conjugate idempotents and unit regular elements modulo an ideal I of a UN ring R. Matrix rings over UN rings are discussed and it is obtained that for a commutative ring R, a matrix ring M_n(R) is UN if and only if R is UN. Lastly, UN group rings are investigated and we obtain the conditions on a group G and a field K for the group algebra KG to be UN. Then we extend the results obtained for KG to the group ring RG over a ring R (which may not necessarily be a field).

• Bulletin of the Korean Mathematical Society
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• v.60 no.2
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• pp.339-348
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• 2023

It is shown that every nilpotent-invariant module can be decomposed into a direct sum of a quasi-injective module and a square-free module that are relatively injective and orthogonal. This paper is also concerned with rings satisfying every cyclic right R-module is nilpotent-invariant. We prove that R ≅ R₁ × R₂, where R₁, R₂ are rings which satisfy R₁ is a semi-simple Artinian ring and R₂ is square-free as a right R₂-module and all idempotents of R₂ is central. The paper concludes with a structure theorem for cyclic nilpotent-invariant right R-modules. Such a module is shown to have isomorphic simple modules eR and fR, where e, f are orthogonal primitive idempotents such that eRf ≠ 0.

• Kyungpook Mathematical Journal
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• v.63 no.2
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• pp.175-186
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• 2023

In this article we look at the property of a 2 by 2 full matrix ring over the ring of integers, of being weakly right IQNN. This generalisation of the property of being right IQNN arises from products of idempotents and nilpotents. We shown that it is, indeed, a proper generalization of right IQNN. We consider the property of beign weakly right IQNN in relation to several kinds of factorizations of a free algebra in two indeterminates over the ring of integers modulo 2.

• Bulletin of the Korean Mathematical Society
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• v.30 no.2
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• pp.187-191
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• 1993

In this note we prove the following result: Let A be a complex Banach *-algebra with continuous involution and let B be an $A^{*}$-algebra./T(A) = B. Then T is continuous (Theorem 2). From above theorem some others results of special interest and some well-known results follow. (Corollaries 3,4,5,6 and 7). We close this note with some generalizations and some remarks (Theorems 8.9.10 and question). Throughout this note we consider only complex algebras. Let A and B be complex algebras. A linear mapping T from A into B is called jordan homomorphism if T( $x^{1}$) = (Tx)$^{2}$ for all x in A. A linear mapping T : A .rarw. B is called spectrally-contractive mapping if .rho.(Tx).leq..rho.(x) for all x in A, where .rho.(x) denotes spectral radius of element x. Any homomorphism algebra is a spectrally-contractive mapping. If A and B are *-algebras, then a homomorphism T : A.rarw.B is called *-homomorphism if (Th)$^{*}$=Th for all self-adjoint element h in A. Recall that a Banach *-algebras is a complex Banach algebra with an involution *. An $A^{*}$-algebra A is a Banach *-algebra having anauxiliary norm vertical bar . vertical bar which satisfies $B^{*}$-condition vertical bar $x^{*}$x vertical bar = vertical bar x vertical ba $r^{2}$(x in A). A Banach *-algebra whose norm is an algebra $B^{*}$-norm is called $B^{*}$-algebra. The *-semi-simple Banach *-algebras and the semi-simple hermitian Banach *-algebras are $A^{*}$-algebras. Also, $A^{*}$-algebras include $B^{*}$-algebras ( $C^{*}$-algebras). Recall that a semi-prime algebra is an algebra without nilpotents two-sided ideals non-zero. The class of semi-prime algebras includes the class of semi-prime algebras and the class of prime algebras. For all concepts and basic facts about Banach algebras we refer to [2] and [8].].er to [2] and [8].].