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Let R be a prime ring with its Utumi ring of quotients U, C = Z(U) extended centroid of R, F a nonzero generalized derivation of R, L a noncentral Lie ideal of R and k ≥ 2 a fixed integer. Suppose that there exists 0 ≠ a ∊ R such that a|F(un1), un2, . . . , unk ] = 0 for all u ∊ L, where n1, n2, . . . , nk ≥ 1 are fixed integers. Then either there exists λ ≥ ∊ C such that F(x) = ≥x for all x ∊ R, or R satisfies s4, the standard identity in four variables. [ABSTRACT FROM AUTHOR]

Let R be an associative ring and σ, τ two epimorphisms of R. A mapping F : R → R (not necessarily additive) is said to be a multiplicative (generalized)-(σ, τ)-derivation if F(xy) = F(x)σ(y) + τ(x)g(y) holds for all x, y ∈ R, where g is any mapping on R. The main object in this article is to study the situations (1) F(xy) - F(x)F(y) = 0, (2) F(xy) + F(x)F(y) = 0, (3) F(xy) - F(y)F(x) = 0 and (4) F(xy) + F(y)F(x) = 0; for all x, y in some nonzero subsets of prime and semiprime rings, where F is a multiplicative (generalized)-(σ, σ)-derivation of R. [ABSTRACT FROM AUTHOR]

This paper is a sequel of our previous paper [1]. In this paper, we introduce the notions of regular ideal and partial ideal (p-ideal) in a ternary semiring and using these two notions we characterize regular ternary semiring. [ABSTRACT FROM AUTHOR]

In this paper, we continue our study of the square element graph which is defined as follows: if R is a ring then the square element graph 핊q(R) is the simple undirected graph whose vertex set consists of all non-zero elements of R, and two vertices u, v are adjacent if and only if u + v = x2 for some x ϵ R - {0}. Here we mainly consider the connectedness of 핊q(R) over different rings. We particularly look at 핊q(R) taken over some infinite rings which contain ℤ as a subring. For a ring R and an ideal I of R, the relation between the connectedness of 핊q(R) and 핊q(R/I) is studied. Finally, connectedness of 핊q(R) over various polynomial rings and matrix rings are considered. [ABSTRACT FROM AUTHOR]