جستجوی مقالات مرتبط با کلیدواژه « Prime » در نشریات گروه « علوم پایه »

In this paper we introduce the concept of ideal elements in poe-AG-groupoid and give some characterizations and properties of their ideal elements. So we consider some results concerning ideals in poe-semigroups and investigate them in poe-AG-groupoids. Also, the class of ideal elements of poe-AG-groupoids are studied, certain intrinsic and basic properties of poe-AG-groupoids including: ideal, bi-ideal, interior ideal, prime, semiprime, intra-regular elements and etc. are studied as well. The corresponding results on poe-semigroups can be also obtained as application of the results of this paper.

Let $R$ be a prime ring with $U$ the Utumi quotient ring and $Q$ be the Martindale quotient ring of $R$, respectively. Let $d$ be a derivation of $R$ and $m,n$ be fixed positive integers. In this paper, we study the case when one of the following holds:$(i)$~ $d(x)circ_n d(y)$=$xcirc _m y$ $(ii)$~$d(x)circ_m d(y)$=$(d(xcirc y))^n$ for all $x,y$ in some appropriate subset of $R$. We also examine the case where $R$ is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on Banach algebras.
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We show that in {\bf ZF} set theory without choice, the Ultrafilter \mbox{Principle} ({\bf UP}) is equivalent to several compactness theorems for Alexandroff discrete spaces and to Rudin's Lemma, a basic tool in topology and the theory of quasi\-continuous domains. Important consequences of Rudin's Lemma are various lift lemmas, saying that certain properties of posets are inherited by the free unital semilattices over them. Some of these principles follow not only from {\bf UP} but also from {\bf DC}, the Principle of Dependent Choices. On the other hand, they imply the Axiom of Choice for countable families of finite sets,which is not provable in ZF
set theory.

Let R be a prime ring with characteristic different from two, I be a nonzero ideal of R, and F be a generalized derivation associated with a nonzero derivation d of R. In the present paper we investigate the commutativity of R satisfying the relation F([x, y]k) n = ([x, y]k) l for all x, y ∈ I, where l, n, k are fixed positive integers. Moreover, let R be a semiprime ring, A = O(R) be an orthogonal completion of R, and B = B(C) be the Boolean ring of C. Suppose F([x, y]k) n = ([x, y]k) l for all x, y ∈ R, then there exists a central idempotent element e of B such that d vanishes identically on eA and the ring (1 − e)A is commutative.

The aim of this short note is to introduce the concepts of prime and semiprime ideals in ordered AG-groupoids with left identity. These concepts are related to the concepts of quasi-prime and quasi-semiprime ideals, play an important role in studying the structure of ordered AG-groupoids, so it seems to be interesting to study them.

Let R be a ∗-prime ring with centerý ýZ(R)ý, ýd a non-zero (σ,τ)-derivation of R with associatedý ýautomorphisms σ and τ of Rý, ýsuch that σý, ýτý ýand d commute with ′∗′ý. ýSuppose that U is an ideal of R such that U∗=Uý, ýand Cσ,τ={c∈ýýR | cσ(x)=τ(x)c for~all x∈R}. In the present paperý, ýit is shown that if characteristic of R is different from two andý ý[d(U),d(U)]σ,τ={0}, then R is commutativeý. ýCommutativity of R has also been established in case ifý ý[d(R),d(R)]σ,τ⊆Cσ,τ.

A polynomial 1 2 (,, ,) n f X X X is called multilinear if it is homogeneous and linear in every one of its variables. In the present paper our objective is to prove the following Let R be a prime K-algebra over a commutative ring K with unity and let 1 2 (,, ,) n f X X X be a multilinear polynomial over K. Suppose that d is a nonzero derivation on R such that 1 2 1 2 (,, ,) (,, ,) s t df x x x n  f x x xn for all 1 2 ,, , n x x x R, where s,t are fixed positive integers. Then 1 2 (,, ,) n f X X X is central-valued on R . We also examine the case R which is a semiprime K-algebra.

The notions of quasi-prime submodules and developed Zariski topology was introduced by the present authors in \cite{ah10}. In this paper we use these notions to define a scheme. For an R-module M, let X:={Q∈q\Spec(M)∣(Q:RM)∈\Spec(R)}. It is proved that (X,OX) is a locally ringed space. We study the morphism of locally ringed spaces induced by R-homomorphism M→N, and also by ring homomorphism R→S. Among other results, we show that (X,OX) is a scheme by putting some suitable conditions on M.

Let M and N be two non-zero right R-modules, M is called subhomomorphic to N in case there exist R−homomorphisms f: M ! N, g : N ! M such that gof is non-zero, and M is called strongly subhomomorphic to N in case there exist homomorphisms f :M! N, g : N ! M such that both fog and gof are non-zero. After establishing some basic properties of (strongly) subhomomorphic to a ring, it is shown that for a nonsingular ring R the class of injective right R -modules are subhomomorphic to R if and only if R is a semisimple ring.