Representation dimensions of triangular matrix algebras

Derived Equivalent Mates of Triangular Matrix Algebras

A triangular matrix algebra over a field k is defined by a triplet (R, S, M) where R and S are k-algebras and RMS is an SR-bimodule. We show that if R, S and M are finite dimensional and the global dimensions of R and S are finite, then the triangular matrix algebra corresponding to (R, S, M) is derived equivalent to the one corresponding to (S, R, DM), where DM = Homk(M, k) is the dual of M , ...

Band Matrix Representation of Triangular InputBalanced

For generic lower triangular matrices, A, we prove that A ij = P d q=1 H iq G jq for i > j is equivalent to A = M ?1 N where M and N are d+1 banded matrices. A lower triangular matrix A is input balanced of order/rank d if there exists a rank-d matrix B such that AA = I ? BB. We prove that if A is triangular input balanced then generically, A = M ?1 N where M and N are d + 1 banded matrices. Th...

Derived dimensions of representation-finite algebras

It is shown that the derived dimension of any representation-finite Artin algebra is at most one. Mathematics Subject Classification 2000: 16G10, 18E30, 18E10. Let T be a triangulated category, and I and J two full subcategories of T . Denote by 〈I〉 the smallest full subcategory of T containing I and closed under shifts, finite direct sums and direct summands. Denote by I∗J the full subcategory...

Biflatness and Biprojectivitiy of Triangular Banach Algebras

NILPOTENT GRAPHS OF MATRIX ALGEBRAS

Let $R$ be a ring with unity. The undirected nilpotent graph of $R$, denoted by $Gamma_N(R)$, is a graph with vertex set ~$Z_N(R)^* = {0neq x in R | xy in N(R) for some y in R^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in N(R)$, or equivalently, $yx in N(R)$, where $N(R)$ denoted the nilpotent elements of $R$. Recently, it has been proved that if $R$ is a left A...