Armendariz And Semicommutative Properties Of Matrix Rings

Let R be an associative ring with identity.A ring R is called Armendariz if, whenever (∑_i=0^m a_ixⁱ)(∑_j=0ⁿb_jx^j) = 0 in R[x], a_ib_j= 0 for all i and j. A ring R is called reduced if it has no non-zero nilpotent elements.In this paper, we do the study of simple 0 - multiplication, Armendariz and semicommutative properties of matrix rings.On simple 0 - multiplication rings, we identify three classes of simple 0 -multiplication subrings of upper triangular matrix rings over reduced ring. Two classes among them are simple 0 - multiplication subrings of upper triangular matrix rings, and the third class among them is a maximal simple 0-multiplication subrings of upper triangular matrix rings.On Armendariz rings, first, we identify five classes of Armendariz subrings of matrix rings over a reduced ring. Two classes among them are Armendariz subringsof upper triangular matrix rings over a reduced ring, and a class among them is a maximal Armendariz subring of upper triangular matrix rings over a reduced ring, and two classes among them are maximal general Armendariz subrings of matrixrings over a reduced ring. Second, identify five classes ofα-skew Armendariz subrings of matrix rings over aα-rigid ring. Two classes among them areα-skew Armendariz subrings of upper triangular matrix rings over aα-rigid ring, and a class among them is a maximalα-skew Armendariz subring of upper triangular matrix rings over aα-rigid ring, and two classes among them are maximal generalα-skew Armendariz subrings of matrix rings over aα-rigid ring. Last, we identify five classes of M-Armendariz subrings of matrix rings over a M-Armendariz and reduced ring. Two classes among them are M-Armendariz subrings of upper triangularmatrix rings over a M-Armendariz and reduced ring, and a class among them is a maximal M-Armendariz subring of upper triangular matrix rings over a M-Armendariz and reduced ring, and two classes among them are maximal general M-Armendariz subrings of matrix rings over a M-Armendariz and reduced ring.On semicommutative rings, we identify five classes of semicommutative subringsof matrix rings over a reduced ring. Two classes among them are semicommuta tive subrings of upper triangular matrix rings over a reduced ring, and a class among them is a maximal semicommutative subring of upper triangular matrix rings over a reduced ring, and two classes among them are maximal general semicommutative subrings of matrix rings over a reduced ring, and a class among them also is a maximal general symmetric subrings of matrix rings over a reduced ring .