Journal of Algebra and Its Applications

We study the category for the queer Lie superalgebra 𝔮(n), and the corresponding block decomposition induced by infinitesimal central characters. In particular, we show that the so-called typical blocks correspond to standardly stratified algebras, in the sense of Cline, Parshall and Scott. By standard arguments for Lie algebras, modified to the superalgebra situation, we prove that these CPS-stratified algebras have finite finitistic dimension and the double centralizer property. Moreover, we prove that certain strongly typical blocks are equivalent. Finally, we generalize Kostant's Theorem to the 𝔮(n)-case and describe all typical 𝔮(2)-blocks.

Let M be a Verma module over the Lie algebra, sl₂(k), of trace zero 2 × 2 matrices over the algebraically closed field k. We show that the ring, R_M, of definable scalars of M is a von Neumann regular ring and that the canonical map from U(sl₂(k)) to R_M is an epimorphism of rings. We also describe the Ziegler closure of M. The proofs make use of ideas from the model theory of modules.

A ring R is called left morphic if R/Ra ≅ l(a) for each a ∈ R, equivalently if there exists b ∈ R such that Ra = l(b) and l(a) = Rb. In this paper, we ask only that b and c exist such that Ra = l(b) and l(a) = Rc, and call R left quasi-morphic if this happens for every element a of R. This class of rings contains the regular rings and the left morphic rings, and it is shown that finite intersections of principal left ideals in such a ring are again principal. It is further proved that if R is quasi-morphic (left and right), then R is a Bézout ring and has the ACC on principal left ideals if and only if it is an artinian principal ideal ring.

Let M be a Clifford monoid and let θ be an endomorphism of M. We prove that if the Bruck–Reilly extension BR(M, θ) is finitely presented then M is finitely generated. This allows us to derive necessary and sufficient conditions for Bruck–Reilly extensions of Clifford monoids to be finitely presented.

Let K, S, D be a division ring, an endomorphism and a S-derivation of K, respectively. In this setting we introduce generalized noncommutative symmetric functions and obtain Viète formula and decompositions of differential operators. W-polynomials show up naturally, their connections with P-independency, Vandermonde and Wronskian matrices are briefly studied. The different linear factorizations of W-polynomials are analyzed. Connections between the existence of LLCM of monic linear polynomials with coefficients in a ring and the left duo property are established at the end of the paper.

We investigate the module theory of a certain class of semilocal rings connected with nearly simple uniserial domains. For instance, we classify finitely presented and pure-projective modules over these rings and calculate their projective dimension.

In this paper, we deal with the study of maximal non-universally catenarian subrings (that is a non-universally catenarian subring R of a domain S such that any subring of S that properly contains R is universally catenarian). Several satisfactory results are given, and for S a field and R not integrally closed, we characterize these domains.

A domain R is a maximal non-ACCP subring of its quotient field if and only if R is either a two-dimensional valuation domain with a DVR overring or a one-dimensional nondiscrete valuation domain. If R ⊂ S is a minimal ring extension and S is a domain, then (R,S) is a residually algebraic pair. If S is a domain but not a field, a maximal non-ACCP subring extension R ⊂ S is a minimal ring extension if (R,S) is a residually algebraic pair and R is quasilocal. Results with a similar flavor are given for domains R ⊂ S sharing a nonzero ideal, with applications to rings R of the form A + XB[X] or A + XB[[X]]. If R ⊂ S is a minimal ring extension such that R is a domain and S is not (R-algebra isomorphic to) an overring of R, then R satisfies ACCP if and only if S satisfies ACCP.