In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule. If T {\displaystyle T} and U...

abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set...

hereditary. If S is a von Neumann regular ring with an ideal I that is not a direct summand, then the triangular matrix ring [ S / I S / I 0 S ] {\displaystyle...

square matrix over a commutative ring is invertible if and only if its determinant is a unit (i.e. multiplicatively invertible element) of that ring. The...

in matrix theory. Square matrices of a given dimension form a noncommutative ring, which is one of the most common examples of a noncommutative ring. The...

both upper- and lower-triangular. A diagonal matrix is symmetric. The identity matrix In and zero matrix are diagonal. A 1×1 matrix is always diagonal....

determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a 2 × 2 matrix is | a b c d | = a d − b c ,...

element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n. In particular, a square matrix M is a unipotent...

or left Kasch. The two-by-two upper (or lower) triangular matrix ring is not right or left Kasch. A ring with right socle zero (i.e. s o c ( R R ) = {...

algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It...