Semiprime ring

Let R be an associative ring. We define a subset S R of R as S R = {a ∈ R | aRa = (0)} and call it the source of semiprimeness of R. We first examine some basic properties of the subset S R in any ring R, and then define the notions such as R being a |S R |-reduced ring, a |S R |-domain and a |S R |-division ring which are slight generalizations of their classical versions. Beside others, we for instance prove that a finite |S R |-domain is necessarily unitary, and is in fact a |S R |-division ring. However, we provide an example showing that a finite |S R |-division ring does not need to be commutative. All possible values for characteristics of unitary |S R |-reduced rings and |S R |-domains are also determined.

In this paper, it is defined that left *-α-derivation, generalized left *-α-derivation and *-α-derivation, generalized *-α-derivation of a *-ring where α is a homomorphism. The results which proved for generalized left *-derivation of R in [1] are extended by using generalized left *-α-derivation. The commutativity of a *-ring with generalized left *-α-derivation is investigated and some results are given for generalized *-α-derivation.

We show the existence of principally (and finitely generated) right FI-extending right ring hulls for semiprime rings. From this result, we prove that right principally quasi-Baer (i.e., right p.q.-Baer) right ring hulls always exist for semiprime rings. This existence of right p.q.-Baer right ring hull for a semiprime ring unifies the result by Burgess and Raphael on the existence of

In this study, we prove that any nonzero reverse (,) − biderivation on a prime ring is (,) − biderivation. Also, we show that any Jordan (,) − biderivation on non-commutative semi-prime ring with ℎ () ≠ 2 is an (,) − biderivation. In addition, we investigate commutative feature of prime ring with Jordan left (,) − biderivation.

The main purpose of this paper is to prove the following result: Let R be a 2-torsion free semiprime *-ring. Suppose that θ, φ are endomorphisms of R such that θ is onto. If there exists an additive mapping F: R → R associated with a (θ, φ)-derivation d of R such that F(xx*) = F(x)θ(x*) + φ(x)d(x*) holds for all x ∈ R, then F is a generalized (θ, φ)-derivation. Further, some more related results are obtained.

In the theory of rings and modules there is a correspondence between certain ideals of a ring R and submodules of an R-module that arise from annihilation. The submodules obtained using annihilation, which correspond to prime ideals play an important role in decomposition theory. In this paper, we attempt to intuitionistic fuzzify the concept of annihilators of subsets of modules. We investigate certain characterization of intuitionistic fuzzy annihilators of subsets of modules. Using the concept of intuitionistic fuzzy annihilators, intuitionistic fuzzy prime submodules and intuitionistic fuzzy annihilator ideals are defined and various related properties are established. KEYWORDS Intuitionistic fuzzy submodule, intuitionistic fuzzy prime submodule, intuitionistic fuzzy ideal, intuitionistic fuzzy annihilator, semiprime ring.

The main purpose of this paper is to prove the following result: Let R be a 2-torsion free semiprime *-ring. Suppose that θ, φ are endomorphisms of R such that θ is onto. If there exists an additive mapping F: R → R associated with a (θ, φ)-derivation d of R such that F(xx*) = F(x)θ(x*) + φ(x)d(x*) holds for all x ∈ R, then F is a generalized (θ, φ)-derivation. Further, some more related results are obtained.

In this paper, we define a set including of all fa with a ∈ R generalized derivations of R and is denoted by f R. It is proved that (i) the mapping g : L (R) → f R given by g (a) = f −a for all a ∈ R is a Lie epimorphism with kernel Nσ,τ ; (ii) if R is a semiprime ring and σ is an epimorphism of R, the mapping h : f R → I (R) given by h (fa) = i σ(−a) is a Lie epimorphism with kernel l (f R) ; (iii) if f R is a prime Lie ring and A, B are Lie ideals of R, then [f A , f B ] = (0) implies that either f A = (0) or f B = (0).

Necessary and sufficient conditions for an Ore extension S = R[x;σ,δ] to be a PI ring are given in the case σ is an injective endomorphism of a semiprime ring R satisfying the ACC on annihilators. Also, for an arbitrary endomorphism τ of R, a characterization of Ore extensions R[x;τ] which are PI rings is given, provided the coefficient ring R is noetherian.

In this paper, it is defined that left *-α-derivation, generalized left *-α-derivation and *-α-derivation, generalized *-α-derivation of a *-ring where α is a homomorphism. The results which proved for generalized left *-derivation of R in [1] are extended by using generalized left *-α-derivation. The commutativity of a *-ring with generalized left *-α-derivation is investigated and some results are given for generalized *-α-derivation.

Let $M$ be a noncommutative 2-torsion free semiprime $\Gamma$-ring satisfying a certain assumption and let $S$ and $T$ be left centralizers on $M$. We prove the following results: \\(i) If $[S(x),T(x)]_{\alpha }\beta S(x)+S(x)\beta [S(x),T(x)]_{\alpha }$=$0$ holds for all $x\in M$ and $\alpha ,\beta \in \Gamma $, then $[S(x),T(x)]_{\alpha }$=$0$. \\(ii) If $S\neq 0 (T\neq 0)$, then there exists $\lambda \in C$,(the extended centroid of $M$) such that $T$=$\lambda \alpha S(S=\lambda \alpha T)$ for all $\alpha \in \Gamma $. \\(iii) Suppose that $[[S(x),T(x)]_{\alpha },S(x)]_{\beta }$=$0$ holds for all $x\in M$ and $\alpha ,\beta \in\Gamma $. Then $[S(x),T(x)]_{\alpha }$=$0$ for all $x\in M$ and $\alpha \in\Gamma $. \\(iv) If $M$ is a prime $\Gamma $-ring satisfying a certain assumption and $S\neq 0(T\neq 0)$, then there exists $\lambda \in C$, the extended centroid, such that $T$=$\lambda \alpha S(S=\lambda \alpha T)$.