Let w be a word in the free group of rank n ∈ N and let V(w) be the variety of groups defined by the law w = 1. Define V(w∗) to be the class of all groups G in which for any infinite subsets X1, . . . , Xn there exist xi ∈ Xi, 1 ≤ i ≤ n, such that w(x1, . . . , xn) = 1. Clearly, V(w) ∪ F ⊆ V(w ∗); F being the class of finite groups. In this paper, we investigate some words w and some certain cl...

If [Formula: see text] is a directed graph and field, the Leavitt path algebra of over naturally graded by group integers We formulate properties which are equivalent with being crossed product, skew ring, or ring respect to this natural grading. state main result so that also characterized in terms pre-ordered Grothendieck text]-group text]. has finitely many vertices, we characterize when str...

Let R be a finite-dimensional algebra over an algebraically closed field F graded by an arbitrary group G. We prove that R is a graded division algebra if and only if it is isomorphic to a twisted group algebra of some finite subgroup of G. If the characteristic of F is zero or char F does not divide the order of any finite subgroup of G then we prove that R is graded simple if and only if it i...

Various topics on affine rings are considered, such as the relationship between Gelfand-Kirillov dimension and Krull dimension, and when a "locally affine" algebra is affine. The dimension result is applied to study prime ideals in fixed rings of finite groups, and in identity components of group-graded rings. 0. Introduction. We study affine rings (i.e., rings finitely generated as algebras ov...

Let G be a group and R G-graded ring. In this paper, we present examine the concept of graded weakly 2-absorbing ideals as in generality prime ring which is not commutative, demonstrates that symmetry obtained lot outcomes commutative rings remain are commutative.

BACKGROUNDS Despite the potential therapeutic benefits, cell therapy in renal ischemia-reperfusion (I/R) injury is currently limited by low rates of cell engraftment after systemic delivery. In this study, we investigate whether locally administration through renal artery can enhance the migration and therapeutic potential of mesenchymal stem cells (MSCs) in ischemic kidney. METHODS The model...