On rings in which every maximal one-sided ideal contains a maximal ideal

Given a ring R, consider the condition: (^*) every maximal right ideal of R contains a maximal ideal of R. We show that, for a ring R and 0 ≠ e ² = e ∈ R such that ele ? eRe every proper ideal I of R R satisfies (^*) if and only if eRe satisfies (^*). Hence with the help of some other results, (^*) is a Morita invariant property. For a simple ring R R[x] satisfies (^*) if and only if R[x] is not right primitive. By this result, if R is a division ring and R[x] satisfies (^*), then the Jacobson conjecture holds. We also show that for a finite centralizing extension S of a ring R R satisfies (^*) if and only if S satisfies (^*).     

On representable linearly compact modules

For a flat module we prove that is a functor from the category of linearly compact modules to itself and is exact. Moreover, is representable when is linearly compact and representable. This gives an affirmative answer to a question of L. Melkersson (1995) for linearly compact modules without the condition of finite Goldie dimension. The set of attached prime ideals of the co-localization of a linearly compact representable module with respect to a multiplicative set in is described.

     

The Golod property of powers of the maximal ideal of a local ring

We identify minimal cases in which a power \(\mathfrak {m}^i\not =0\) of the maximal ideal of a local ring R is not Golod, i.e. the quotient ring \(R/\mathfrak {m}^i\) is not Golod. Complementary to a 2014 result by Rossi and ?ega, we prove that for a generic artinian Gorenstein local ring with \(\mathfrak {m}^4=0\ne \mathfrak {m}^3\), the quotient \(R/\mathfrak {m}^3\) is not Golod. This is provided that \(\mathfrak {m}\) is minimally generated by at least 3 elements. Indeed, we show that if \(\mathfrak {m}\) is 2-generated, then every power \(\mathfrak {m}^i\ne 0\) is Golod.     

On the regular representation of a nonunimodular locally compact group

We study the discrete part of the regular representation of a locally compact group and also its Type I part if the group is separable. Our results extend to nonunimodular groups' known results for unimodular groups about formal degrees of square integrable representations, and the Plancherel formula. We establish orthogonality relations for matrix coefficients of square integrable representations and we show that the formal degree in general is not a positive number, but a positive self-adjoint unbounded operator, semi-invariant under the representation. Integrable representations are also studied in this context. Finally we show that when the group is nonunimodular, “Plancherel measure” is not a true measure, but a measure multiplied by a section of a certain real oriented line bundle on the dual space of the group.     

The Poincaré series of a simple complete ideal of a two-dimensional regular local ring

The aim of this work is to introduce both a classical and a motivic Poincaré series associated with a residually rational simple complete m-primary ideal ℘ of a two-dimensional regular local ring (R,m). We describe them in terms of the generators of the value semigroup of ℘, and compare them with the Poincaré series arising from a general element f for ℘.     

On the left regular representation of a separable locally compact group

For an arbitrary separable locally compact group G we exhibit a canonical Borel subset $\text{G}\text{?}{}_{\mathrm{\Delta }}$ of the quasi-dual $\text{G}\text{?}\text{of}\text{G}$ (with the Mackey Borel structure), such that $\text{G}\text{?}{}_{\mathrm{\Delta }}$ is a standard Borel space in the induced Borel structure, and such that the canonical measure for the left regular representation λ_Gof G is concentrated on $\text{G}\text{?}{}_{\mathrm{\Delta }}$. On the basis of this we discuss the (non-unimodular) “Plancherel theorem.”     

On some algebraic properties of locally compact and weakly linearly compact topological Abelian groups

Locally compact and weakly linearly compact topological groups are studied. The notion of a weakly linearly compact topological Abelian group is a generalization of the notion of a weakly separable topological Abelian group, introduced by N. Ya. Vilenkin. Some algebraic properties of these groups are studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 41, Topology and Its Applications, 2006.