associative ring
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2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Aslam Siddeeque ◽  
Nazim Khan

Abstract Let R be an associative ring. A multiplicative semi-derivation d is a map on R satisfying d ⁢ ( x ⁢ y ) = d ⁢ ( x ) ⁢ g ⁢ ( y ) + x ⁢ d ⁢ ( y ) = d ⁢ ( x ) ⁢ y + g ⁢ ( x ) ⁢ d ⁢ ( y )   and   d ⁢ ( g ⁢ ( x ) ) = g ⁢ ( d ⁢ ( x ) ) {d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y)\quad\text{and}\quad d(g(x))=g(d(x))} for all x , y ∈ R {x,y\in R} , where g is any map on R. In this paper, we have obtained some conditions on R, which make d additive. Finally, we have also shown that every multiplicative semi-derivation on M n ⁢ ( ℂ ) {M_{n}(\mathbb{C})} , the algebra of all n × n {n\times n} matrices over the field ℂ {\mathbb{C}} of complex numbers, is an additive derivation.


2021 ◽  
Vol 26 (4) ◽  
Author(s):  
Ikram Saed

Let R be an associative ring with center Z(R) , I be a nonzero ideal of R and  be an automorphism  of R . An 3-additive mapping M:RxRxR R is called a symmetric left -3-centralizer if M(u1y,u2 ,u3)=M(u1,u2,u3)(y) holds for all  y, u1, u2, u3 R . In this paper , we shall investigate the  commutativity of prime rings admitting symmetric left -3-centralizer satisfying any one of the following conditions : (i)M([u ,y], u2, u3)  [(u), (y)] = 0 (ii)M((u ∘ y), u2, u3)  ((u) ∘ (y)) = 0 (iii)M(u2, u2, u3)  (u2) = 0 (iv) M(uy, u2, u3)  (uy) = 0 (v) M(uy, u2, u3)  (uy) For all u2,u3 R and u ,y I


Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1909
Author(s):  
Małgorzata Jastrzębska

The present paper is part of the research on the description of rings with a given property of the lattice of left (right) annihilators. The anti-isomorphism of lattices of left and right annihilators in any ring gives some kind of symmetry: the lattice of left annihilators is Boolean (complemented, distributive) if and only if the lattice of right annihilators is such. This allows us to restrict our investigations mainly to the left side. For a unital associative ring R, we prove that the lattice of left annihilators in R is Boolean if and only if R is a reduced ring. We also prove that the lattice of left annihilators of R being two-sided ideals is complemented if and only if this lattice is Boolean. The last statement, in turn, is known to be equivalent to the semiprimeness of R. On the other hand, for any complete lattice L, we construct a nilpotent ring whose lattice of left annihilators coincides with its sublattice of left annihilators being two-sided ideals and is isomorphic to L. This construction shows that the assumption of R being unital cannot be dropped in any of the above two results. Some additional results on rings with distributive or complemented lattices of left annihilators are obtained.


2021 ◽  
Author(s):  
Stuart D. Scott

Binary groups are a meaningful step up from non-associative rings and nearrings. It makes sense to study them in terms of their nearrings of zero-fixing polynomial maps. As this involves algebras of a more specialized nature these are looked into in sections three and four. One of the main theorems of this paper occurs in section five where it is shown that a binary group V is a P0(V) ring module if, and only if, it is a rather restricted form of non-associative ring. Properties of these non-associative rings (called terminal rings) are investigated in sections six and seven. The finite case is of special interest since here terminal rings of odd order really are quite restricted. Sections eight to thirteen are taken up with the study of terminal rings of order pn (p an odd prime and n ≥ 1 an integer ≤ 7).


2021 ◽  
Vol 65 (3) ◽  
pp. 38-45
Author(s):  
Ayazul Hasan

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. In this paper, we study the existence of several classes C of QTAG-modules which satisfy the property that M belongs to C uniquely when M/N belongs to C provided that N is a finitely generated submodule of the QTAG-module.


Author(s):  
Peter V. Danchev ◽  
Tsiu-Kwen Lee

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.


2021 ◽  
pp. 2357-2361
Author(s):  
Alaa A. Elewi

Let be an associative ring with identity and let be a unitary left -module. Let  be a non-zero submodule of .We say that  is a semi- - hollow module if for every submodule  of  such that  is a semi- - small submodule ( ). In addition, we say that  is a semi- - lifting module if for every submodule  of , there exists a direct summand  of  and  such that   The main purpose of this work was to develop the properties of these classes of module.  


Author(s):  
A. S. Monastyreva

In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we find all graphs containing four vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring.


2021 ◽  
Vol 31 (3) ◽  
pp. 223-230
Author(s):  
Alexey D. Yashunsky

Abstract We consider the transformations of random variables over a finite associative ring by the addition and multiplication operations. For arbitrary finite rings, we construct families of distribution algebras, which are sets of distributions closed over sums and products of independent random variables.


2021 ◽  
pp. 1271-1275
Author(s):  
Eman Mohammed ◽  
Wasan Khalid

Let  R be an associative ring with identity and let M be a left R-module . As a generalization of µ-semiregular modules, we introduce an F-µ-semiregular module. Let F be a submodule of M and x∊M. x is called F-µ-semiregular element in M , if there exists a decomposition M=A⨁B, such that A is a projective submodule of  and . M is called  F-µ-semiregular if x is F-µ-semiregular element for each x∊M. A condition under which the module µ-semiregular is F-µ-semiregular module was given. The basic properties and some characterizations of the F-µ-semiregular module were provided.


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