Additive Maps on Units of Rings

2018 ◽  
Vol 61 (1) ◽  
pp. 130-141
Author(s):  
Tamer Košan ◽  
Serap Sahinkaya ◽  
Yiqiang Zhou
Keyword(s):  
Recent Result ◽  
Direct Product ◽  
Homomorphic Image ◽  
Semilocal Ring ◽  
Additive Map ◽  
Exchange Rings ◽  
Additive Maps

AbstractLet R be a ring. A map f: R → R is additive if f(a + b) = f(a) + f(b) for all elements a and b of R. Here, a map f: R → R is called unit-additive if f(u + v) = f(u) + f(v) for all units u and v of R. Motivated by a recent result of Xu, Pei and Yi showing that, for any field F, every unit-additive map of (F) is additive for all n ≥ z, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring R is additive if and only if either R has no homomorphic image isomorphic to or R/J(R) ≅ with 2 = 0 in R. Consequently, for any semilocal ring R, every unit-additive map of (R) is additive for all n ≥ 2. These results are further extended to rings R such that R/J(R) is a direct product of exchange rings with primitive factors Artinian. A unit-additive map f of a ring R is called unithomomorphic if f(uv) = f(u)f(v) for all units u, v of R. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

scholarly journals Regular semigroups, fundamental semigroups and groups

1980 ◽  
Vol 29 (4) ◽  
pp. 475-503 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Regular Semigroup ◽  
Partial Order ◽  
Direct Product ◽  
Homomorphic Image ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Natural Partial Order ◽  
Regular Semigroups ◽  
Necessary And Sufficient ◽  
Fundamental Semigroups

AbstractIn this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

Additive maps having a generalized derivation expansion

2015 ◽  
Vol 14 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Generalized Derivation ◽  
Elementary Operator ◽  
Additive Map ◽  
Finite Dimensional ◽  
Left And Right ◽  
Central Simple ◽  
Skolem Theorem ◽  
Additive Maps

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.

scholarly journals Additive Maps of Rank k Bivectors

2021 ◽  
Vol 36 (36) ◽  
pp. 847-856
Author(s):  
Wai Leong Chooi ◽  
Kiam Heong Kwa
Keyword(s):  
Linear Spaces ◽  
Exterior Power ◽  
Additive Map ◽  
Additive Maps

Let ${\cal U}$ and ${\cal V}$ be linear spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively, such that Dim$\,{\cal U}=n\geqslant 2$ and $\left|\mathbb{F}\right|\geqslant 3$. Let $\bigwedge^2{\cal U}$ be the second exterior power of ${\cal U}$. Fixing an even integer $k$ satisfying $\frac{n-1}{2}\leqslant k\leqslant n$, it is shown that a map $\psi:\bigwedge^2{\cal U}\rightarrow\bigwedge^2{\cal V}$ satisfies $\psi(u+v)=\psi(u)+\psi(v)$ for all rank $k$ bivectors $u,v\in\bigwedge^2{\cal U}$ if and only if $\psi$ is an additive map. Examples showing the indispensability of the assumption on $k$ are given.

scholarly journals Product structure in topological algebras

1975 ◽  
Vol 20 (4) ◽  
pp. 385-393
Author(s):  
Desmond A. Robbie
Keyword(s):  
Direct Product ◽  
Homomorphic Image ◽  
Product Structure ◽  
Topological Algebras ◽  
Free Semigroups

It is shown that every compact nonconnected semigroup (semiring) which has commuting congruences, has a nontrivial continous homomorphic image which is iseomorphic to a direct product of finite congruence free semigroups (semirings). (This extends parts of earlier work by Kaplansky (1947) on compact rings.) It is also shown that there is a possibly finer representation but onto a product of congruence free semigroups (semirings) known only to be compact Hausdorff. A number of the techniques used evolve from work of Professor Wallace, who retired in mid-1973, and to whom this paper is dedicated.

Commutative rings with infinitely many maximal subrings

2014 ◽  
Vol 13 (07) ◽  
pp. 1450037 ◽  
Author(s):  
Alborz Azarang ◽  
Greg Oman
Keyword(s):  
Direct Product ◽  
Noetherian Ring ◽  
Artinian Ring ◽  
Commutative Rings ◽  
Semilocal Ring ◽  
Finitely Generated ◽  
Fixed Ring ◽  
Irreducible Elements ◽  
Semisimple Ring ◽  
Nonzero Characteristic

It is shown that RgMax (R) is infinite for certain commutative rings, where RgMax (R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then | RgMax (R)| ≥ | Irr (D) ∩ U(R)|, where Irr (D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or | RgMax (R)| ≥ ℵ0. It is proved that if R is a zero-dimensional (or semilocal) ring with | RgMax (R)| < ℵ0, then R has nonzero characteristic, say n, and R is integral over ℤn. In particular, it is shown that if R is an uncountable artinian ring, then | RgMax (R)| ≥ |R|. It is observed that if R is a noetherian ring with |R| > 2ℵ0, then | RgMax (R)| ≥ 2ℵ0. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R2 ⊂ R1 ⊂ R0 = R where each Ri is a maximal subring of Ri-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

scholarly journals On 𝓠-regular semigroups

2018 ◽  
Vol 16 (1) ◽  
pp. 522-530
Author(s):  
Xinyang Feng
Keyword(s):  
Direct Product ◽  
Homomorphic Image ◽  
Regular Semigroups ◽  
Subdirect Products ◽  
Maximum Group

AbstractIn this paper, we give some characterizations of 𝓠-regular semigroups and show that the class of 𝓠-regular semigroups is closed under the direct product and homomorphic images. Furthermore, we characterize the 𝓠-subdirect products of this class of semigroups and study the E-unitary 𝓠-regular covers for 𝓠-regular semigroups, in particular for those whose maximum group homomorphic image is a given group. As an application of these results, we claim that the similar results on V-regular semigroups also hold.

scholarly journals A note on commuting additive maps on rank k symmetric matrices

2021 ◽  
Vol 37 ◽  
pp. 734-746
Author(s):  
Wai Leong Chooi ◽  
Yean Nee Tan
Keyword(s):  
Linear Space ◽  
Symmetric Matrices ◽  
Identity Matrix ◽  
Additive Map ◽  
Additive Maps

Let $n\geq 2$ and $1<k\leq n$ be integers. Let $S_n(\mathbb{F})$ be the linear space of $n\times n$ symmetric matrices over a field $\mathbb{F}$ of characteristic not two. In this note, we prove that an additive map $\psi:S_n(\mathbb{F})\rightarrow S_n(\mathbb{F})$ satisfies $\psi(A)A=A\psi(A)$ for all rank $k$ matrices $A\in S_n(\mathbb{F})$ if and only if there exists a scalar $\lambda\in \mathbb{F}$ and an additive map $\mu:S_n(\mathbb{F})\rightarrow \mathbb{F}$ such that\[\psi(A)=\lambda A+\mu(A)I_n,\]for all $A\in S_n(\mathbb{F})$, where $I_n$ is the identity matrix. Examples showing the indispensability of assumptions on the integer $k>1$ and the underlying field $\mathbb{F}$ of characteristic not two are included.

scholarly journals Subdirect products of E–inversive semigroups

1990 ◽  
Vol 48 (1) ◽  
pp. 66-78 ◽  
Author(s):  
H. Mitsch
Keyword(s):  
Explicit Form ◽  
Direct Product ◽  
Homomorphic Image ◽  
Inverse Semigroups ◽  
Group Congruence ◽  
Subdirect Products ◽  
Special Case ◽  
Maximum Group

AbstractA semigroup S is called E-inversive if for every a ∈ S ther is an x ∈ S such that (ax)2 = ax. A construction of all E-inversive subdirect products of two E-inversive semigroups is given using the concept of subhomomorphism introduced by McAlister and Reilly for inverse semigroups. As an application, E-unitary covers for an E-inversive semigroup are found, in particular for those whose maximum group homomorphic image is a given group. For this purpose, the explicit form of the least group congruence on an arbitrary E-inversive semigroup is given. The special case of full subdirect products of a semilattice and a group (that is, containing all indempotents of the direct product) is investigated and, following an idea of Petrich, a construction of all these semigroups is provided. Finally, all periodic semigroups which are subdirect products of a semilattice or a band with a group are characterized.

scholarly journals On additive maps of prime rings

1995 ◽  
Vol 51 (3) ◽  
pp. 377-381 ◽  
Author(s):  
Matej Brešar ◽  
Bojan Hvala
Keyword(s):  
Prime Ring ◽  
Extended Centroid ◽  
Prime Rings ◽  
Additive Map ◽  
Additive Maps

Let R be a prime ring of characteristic not 2, C be the extended centroid of R, and f: R → R be an additive map. Suppose that [f(x), x2] = 0 for all x ∈ R. Then there exist λ ∈ C and an additive map ζ: R → C such that f(x) = λx + ζ(x) for all x ∈ R. In particular, if f(x)2 = x2 for all x ∈ R, then ζ = 0 and either λ = 1 or λ= -1.

scholarly journals m-commuting Additive Maps on Upper Triangular Matrices Rings

2019 ◽  
pp. 505-514
Author(s):  
Driss Aiat Hadj Ahmed
Keyword(s):  
Commutative Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Additive Map ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Additive Maps ◽  
Upper Triangular Matrix Ring

Let $T_{n}(R)$ be the upper triangular matrix ring over a unital commutative ring whose characteristic is not a divisor of $m$. Suppose that $f:T_{n}(R)\rightarrow T_{n}(R)$ is an additive map such that $X^{m}f(X)=f(X)X^{m}$ for all $x \in T_{n}(R),$ where $m\geq 1$ is an integer. We consider the problem of describing the form of the map $X \rightarrow f(X)$.

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