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...

We characterize the pseudo-equivalence of a block lower triangular matrix T = [Tij ] over a regular ring, and its block diagonal matrix D(T ) = [Tii], in terms of suitable Roth consistency conditions. The latter can in turn be expressed in terms of the solvability of certain matrix equations of the form TiiX − Y Tjj = Uij .

The pseudo-equivalence of a block lower triangular matrix T = [Tij ] over a regular ring and its block diagonal matrix D(T ) = [Tii] is characterized in terms of suitable Roth consistency conditions. The latter can in turn be expressed in terms of the solvability of certain matrix equations of the form TiiX − Y Tjj = Uij .

It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variab...

In this note we prove that elementary surjective maps on triangular algebras are automatically additive. The study of elementary maps was initiated by Brešar and Šerml. Following ([1]), elementary maps are defined as follows. Definition 1. Let R and R be two rings. Suppose that M : R → R and M : R → R are two maps. Call the ordered pair (M,M) an elementary map of R×R if

We study the BGG Category O over a skew group ring, involving a finite group acting on a regular triangular algebra. We relate the representation theory of the algebra to Clifford theory for the skew group ring, and obtain results on block decomposition, semisimplicity, and enough projectives. O is also shown to be a highest weight category; the BGG Reciprocity formula is slightly different bec...

A triangular matrix ring Λ is defined by a triplet (R, S, M) where R and S are rings and RMS is an S-R-bimodule. In the main theorem of this paper we show that if TS is a tilting S-module, then under certain homological conditions on the S-module MS , one can extend TS to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M ...

We give a condition ensuring that the operators in a nilpotent Lie algebra of linear operators on a finite dimensional vector space have a common eigenvector. Introduction Throughout this paper V is a vector space of positive dimension over a field f and g is a nilpotent Lie algebra over f of linear operators on V . An element u ∈ V is an eigenvector for S ⊂ g if u is an eigenvector for every o...

We study certain Schur functors which preserve singularity categories of rings and we apply them to study the singularity category of triangular matrix rings. In particular, combining these results with Buchweitz–Happel’s theorem, we can describe singularity categories of certain non-Gorenstein rings via the stable category of maximal Cohen–Macaulay modules. Three concrete examples of finitedim...