let r be a commutative ring with identity. let n and k be two submodules of a multiplication r-module m. thenn=im and k=jm for some ideals i and j of r. the product of n and k denoted by nk is defined by nk=ijm. inthis paper we characterize some particular cases of multiplication modules by using the product of submodules.

In an earlier paper [S] we have determined the structure of the linear groups over a local ring. In this note we continue the study of the classical groups over a local ring with the investigation of the orthogonal groups. Our main result (cf. Theorem 6 below) is a complete description of the invariant subgroups of an orthogonal group of noncompact type (i.e., of index ^ 1) over a local ring L ...

J. R. Tolkien is admitted as the father of modern fantasy literature. What often overlooked about him that he was a great intellectual, hoping to find an answer issues his time, at least in field which very good: Tolkien`s attitude Evil consists two parts terms Ring. The first traditional Augustinian, later Boethian view. According this view, evil itself nothing. It absence good. So it internal...

Functional testing, as opposed to parametric testing, plays an important role in testing VLSI integrated circuits. However, it appears that designs are not always carefully analysed in advance to determine precisely which faults are clean, i.e. testable by logic means alone. The programmable logic array (PLA) is a popular circuit form used to implement a system of Boolean functions over a set o...

A ring R is called a right Ikeda-Nakayama (for short IN-ring) if the left annihilator of the intersection of any two right ideals is the sum of the left annihilators, that is, if (I ∩ J) = (I) + (J) for all right ideals I and J of R. R is called Armendariz ring if whenever polynomials f (x) = a0 + a1x + ··· + amx, g(x) = b0 + b1x + ··· + bnx ∈ R[x] satisfy f (x)g(x) = 0, then aibj = 0 for each ...

In this paper, we study the transfer of notion $m$-clean ring (i.e., a in which every element is sum unit and an $m$-potent element) to amalgamated bi-amalgamated rings. We also find many sufficient necessary conditions for m-clean property Received: March 7, 2023Accepted: May 31, 2023

 Let $R=oplus_{nin Bbb N_0}R_n$ be a Noetherian homogeneous ring with local base ring $(R_0,frak{m}_0)$, $M$ and $N$ two finitely generated graded $R$-modules. Let $t$ be the least integer such that $H^t_{R_+}(M,N)$ is not minimax. We prove that $H^j_{frak{m}_0R}(H^t_{R_+}(M,N))$ is Artinian for $j=0,1$. Also, we show that if ${rm cd}(R_{+},M,N)=2$ and $tin Bbb N_0$, then $H^t_{frak{m}_0R}(H^2_...

1-Carboxymethyl-3-methylimidazolium chloride {[cmmim]Cl} is utilized as a highly efficient and reusable catalyst for the synthesis of bis(indolyl)methanes by the condensation reaction of indole with arylaldehydes under solvent-free conditions at room temperature. In summary, the presented method have several advantages including simplicity, low cost, generality, efficiency, clean reaction profi...

The author proves that, if S is an FIC-semigroup or a completely regular semigroup, and if RS is a ring with identity, then R < E(S) > is a ring with identity. Throughout this paper, R denotes as a ring with identity. Let S be a semigroup, X ⊆ S . The following notations are used in the paper: < X > : the subsemigroup of S generated by X ; |X| : the cardinal number of X ; E(S): the set of idemp...

1-carboxymethyl-3-methylimidazolium chloride {[cmmim]cl} is utilized as a highly efficient and reusable catalyst for the synthesis of bis(indolyl)methanes by the condensation reaction of indole with arylaldehydes under solvent-free conditions at room temperature. in summary, the presented method have several advantages including simplicity, low cost, generality, efficiency, clean reaction profi...