Leila Heidari Zadeh

• PhD in algebra
• Islamic Azad University, Shoushtar Branch

Let S be a ring with identity in which 2 is invertible. In this paper we describe the structure of the quaternion ring R = H(S) which is a generalization of the Hamilton’s division ring of real quaternions H = H(R).

The category of all $k$-algebras with a bilinear form, whose objects are all pairs $(R,b)$ where $R$ is a $k$-algebra and $b\colon R\times R\to k$ is a bilinear mapping, is equivalent to the category of unital $k$-algebras $A$ for which the canonical homomorphism $(k,1)\to(A,1_A)$ of unital $k$-algebras is a splitting monomorphism in the category o...

Let S be a unital ring in which 2 is invertible, and let \(R=H(S)\) be the quaternion ring over S. In this paper, we characterize the generalized derivations of R and show that every generalized Jordan derivation on R is a generalized derivation. We also consider the question when a derivation (generalized derivation) of a quaternion ring is an inn...

Several elementary properties of the symmetric group Sn extend in a nice way to the full transformation monoid Mn of all maps of the set X := {1, 2, 3,…,n} into itself. The group Sn turns out to be the torsion part of the monoid Mn. That is, there is a pretorsion theory in the category of all maps f:X → X, X an arbitrary finite set, in which biject...

We associate to any ring R with identity a partially ordered set Hom(R), whose elements are all pairs (a,M), where a=ker⁡φ and M=φ−1(U(S)) for some ring morphism φ of R into an arbitrary ring S. Here U(S) denotes the group of units of S. The assignment R↦Hom(R) turns out to be a contravariant functor of the category Ring of associative rings with i...

Several elementary properties of the symmetric group $S_n$ extend in a nice way to the full transformation monoid $M_n$ of all maps of the set $X:=\{1,2,3,\dots,n\}$ into itself. The group $S_n$ turns out to be in some sense the torsion part of the monoid $M_n$. More precisely, there is a pretorsion theory in the category of all maps $f\colon X\to...

Let S be a unital ring in which 2 is invertible, and let R=H(S) be the quaternion ring over S. In this paper, we describe the Lie derivations and generalized Lie derivations of R, we show that if S is commutative or semiprime, then every Lie derivation (resp. generalized Lie derivation) of R decomposes into the sum of a derivation (resp. generalize...

We associate to any ring $R$ with identity a partially ordered set Hom$(R)$, whose elements are all pairs $(\mathfrak a,M)$, where $\mathfrak a=\ker\varphi$ and $M=\varphi^{-1}(U(S))$ for some ring morphism $\varphi$ of $R$ into an arbitrary ring $S$. Here $U(S)$ denotes the group of units of $S$. The assignment $R\mapsto{}$Hom$(R)$ turns out to be...