Let $T=\biggl(\begin{matrix} A&0\\U&B\end{matrix}\biggr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings $U$ is $(B, A)$-bimodule. We first give some computing formulas of projective, injective, flat $FP$-injective dimensions special left $T$-modules. Then we establish (weak) global $T$. It proven that (1) If $U_{A}$ $_{B}U$ $lD(A)\neq lD(B)$, then $lD(T)={\rm m...

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x1,1, . . . , xn,n] to construct irreducible Sn-modules. This construction produces exactly the same matrices as the Kazhdan-Lusztig construction [Invent.Math 53 (1979)], but does not employ the Kazhdan-Lusztig preorders. It also produces exactly the same modules as those which Clausen constructed using a different basis ...

Let R be a noncommutative ring. The zero-divisor graph of R, denoted by Γ(R), is the (directed) graph with vertices Z(R)∗ = Z(R)− {0}, the set of nonzero zero-divisors of R, and for distinct x, y ∈ Z(R)∗, there is an edge x → y if and only if xy = 0. In this paper we investigate the zero-divisor graph of triangular matrix rings over commutative rings. Mathematics Subject Classification: 16S70; ...

let a; b; k 2 k and u ; v 2 u(k). we show for any idempotent e 2 k, ( a 0 b 0 ) is e-clean i ( a 0 u(vb + ka) 0 ) is e-clean and if ( a 0 b 0 ) is 0-clean, ( ua 0 u(vb + ka) 0 ) is too.

In this paper, we introduce the concept of the generalized AIP-rings as a generalization of the generalized quasiBaer rings and generalized p.p.-rings. We show that the class of the generalized AIP-rings is closed under direct products and Morita invariance. We also characterize the 2-by-2 formal upper triangular matrix rings of this new class of rings. Finally, we provide several examples to s...

We introduce a generalization of the Parikh mapping called the Parikh q-matrix encoding, which takes its values in matrices with polynomial entries. The encoding represents a word w over a k-letter alphabet as a (k + 1)-dimensional upper-triangular matrix with entries that are nonnegative integral polynomials in variable q. Putting q = 1, we obtain the morphism introduced by Mateescu, Salomaa, ...