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, ...

Fixed any term-ordering < on the set T(n) of terms in n variables x1, . . . , xn, we study homogeneous ideal a ⊆ P(n) of the ring of polynomials in n variables over a field k, via the associated order-ideal N (a), consisting of all the terms which are not maximal terms of elements of a and called sous-éscalier of a [7, 12, 14]. In particular we will focus our attention on the following combinat...

‎let $mathcal a$ and $mathcal b$ be unital rings‎, ‎and $mathcal m$‎ ‎be an $(mathcal a‎, ‎mathcal b)$-bimodule‎, ‎which is faithful as a‎ ‎left $mathcal a$-module and also as a right $mathcal b$-module‎. ‎let ${mathcal u}=mbox{rm tri}(mathcal a‎, ‎mathcal m‎, ‎mathcal‎ ‎b)$ be the triangular ring and ${mathcal z}({mathcal u})$ its‎ ‎center‎. ‎assume that $f:{mathcal u}rightarrow{mathcal u}$ is...

In this paper, we consider the integrable extended complex modified Korteweg–de Vries equation. Based on Darboux transformation, obtain soliton molecules, positon solutions, rational solutions and rogue waves for Further, under standard decomposition, divide into three patterns: fundamental pattern, triangular pattern ring pattern. On basis of define length width discuss effect different parame...

We introduce Backstr\"om pairs and rings, study their derived categories construct for them a sort of categorical resolutions. For the latter we define global dimension, semi-orthogonal decomposition category deduce that dimension ring is at most $2$. Using this decomposition, description module as elements special bimodule. also partial tilting pair to triangular matrices latter.

The classical theorem of Weitzenböck states that the algebra of invariants K[X] of a single unipotent transformation g ∈ GLm(K) acting on the polynomial algebra K[X] = K[x1, . . . , xm] over a field K of characteristic 0 is finitely generated. This algebra coincides with the algebra of constants K[X] of a linear locally nilpotent derivation δ of K[X]. Recently the author and C. K. Gupta have st...

Associating to each pre-order on the indices 1; :::;n the corresponding structural matrix ring, or incidence algebra, embeds the lattice of n-element pre-orders into the lattice of n n matrix rings. Rings within the order-convex hull of the embedding, i.e. matrix rings that contain the ring of diagonal matrices, can be viewed as incidence algebras of ideal-valued, generalized pre-order relation...

Ž . For a ring R with identity, define Unip R to be the group of upper-triangular n matrices over R all of whose diagonal entries are 1. For i 1, 2, . . . , n 1, let Si denote the matrix whose only nonzero off-diagonal entry is a 1 in the ith row and Ž . Ž . i 1 st column. Then for any integer m including m 0 , it is easy to see that Ž . the S generate Unip Z mZ . Reiner gave relations among th...

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; ...