Showing papers in "Communications in Algebra in 1976"
TL;DR: In this article, coalgebras and cartesian categories are used to define coalgebraic categories in algebraic data. But they do not specify the coalgebraicity of Cartesian categories.
Abstract: (1976). Coalgebras and cartesian categories. Communications in Algebra: Vol. 4, No. 7, pp. 665-667.
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TL;DR: In this article, almost split sequences for integral group rings and orders are presented, where almost split sequence for integral groups and orders is used to define the order of the integral group ring.
Abstract: (1976). Almost split sequences for integral group rings and orders. Communications in Algebra: Vol. 4, No. 10, pp. 893-917.
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TL;DR: In this article, it was shown that under certain chain conditions the Jacobson radical of a polynomial ring $R$(x,δ) consists precisely of polynomials over the nilpotent radical of $R$.
Abstract: A well known result on polynomial rings states that, for a given ring $R$, if $R$ has no non-zero nil ideals then the polynomial ring $R$(x) is semiprimitive, see for example (5) p.12. In this note we study Ore extensions of the form $R$(x,δ), where δ is an automorphism on the ring $R$, with the aim of relating the question of the semiprimitivity of $R$(x,δ) to the presence of non-zero nil ideals in $R$. In particular we show that under certain chain conditions the Jacobson radical of $R$(x,δ) consists precisely of polynomials over the nilpotent radical of $R$. Without restriction on $R$ we show that if δ has finite order then $R$(x,δ) is semiprimitive if $R$ has no nil ideals. Some conditions are required on $R$ and δ for results of the above nature to be true, as illustrated in §5 by an example of a semiprimitive ring $R$ having an automorphism δ of infinite order such that $R$(x,δ) has nil ideals.
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TL;DR: In this article, a Diophantine equation related to finite groups is discussed. But the authors focus on finite groups and do not consider finite groups in the Diophantas.
Abstract: (1976). Diophantine equations related to finite groups. Communications in Algebra: Vol. 4, No. 1, pp. 77-100.
TL;DR: In this article, the cancellation of quasi-injective modules is discussed and discussed in the context of algebraic cancellation of modules, and the authors propose an approach to cancelation of these modules.
Abstract: (1976). On the cancellation of quasi-injective modules. Communications in Algebra: Vol. 4, No. 2, pp. 101-109.
TL;DR: In this paper, the decomposition of gelfand-graev characters of gl3(q) Communications in Algebra: Vol 4, No. 4, pp. 375-401.
Abstract: (1976). Decomposition of gelfand-graev characters of gl3(q) Communications in Algebra: Vol. 4, No. 4, pp. 375-401.
TL;DR: In this paper, the S-good property of group rings and skew polynomial rings was investigated and the results were then applied to group ring AGs with invariance of Tunder ring auto-morphisms.
Abstract: Let ϕ:R→S be a ring homomorphism, and let ϕ∗:Mod-S -> Mod-R be the “restriction of scalars” functor. If Tis a hereditary torsion class for R, then T ∗={MeMod-S ≤ϕ∗(M)eTis a hereditary torsion class for S. We compare the quotient functors Qand Q∗of Tand T ∗respectively. Let ϕ∗:Mod-R→ Mod-S be the “extension of scalars ” functor:ϕ∗(M)=M⊗RS ∀MeMod-R. If ϕ∗ Q Q ∗is called S-good. If Tis S-good, then ϕ∗Q∗is a subfunctor of Qϕ∗If Sis also left R-flat, then ϕ∗Q∗≌,Qϕ ∗.We make an extensive investigation of the S-good property and give a number of ex-amples. The results are then applied to group rings and skew polynomial rings, where the invariance of Tunder ring auto-morphisms plays an essential role. In particular, for a group ring AG, if H◃ Gand Tis a hereditary torsion class for AH which is G-invariant, then Tis AG-good, and with equality if and only if G:H< a or G:H=aand Qcommutes with all free sums. In the special case when the right singular ideal of AHis zero and Tis the Lambek tor-sion class, the second c...
TL;DR: In this paper, it was shown that every cyclic submodule of a uniform quasi-injective right R-module is strongly prime if and only if it is compressible.
Abstract: Let Q be a uniform quasi-injective right R-module Then Q has no proper quasi-injective submodules ≠ 0 if and only if every cyclic submodule is strongly prime. For certain rings a stronger condition is shown to hold, namely that every cyclic submodule is compressible.
TL;DR: In this article, the authors generalize the notion of "descent" of kernel functors for Azumaya algebras to finitely generated Noetherian rings and describe a class of rings in terms of the Zariski topology on Spec.N.b.l.
Abstract: This paper generalizes properties which hold for localization of Azumaya algebras, in two directions. Firstly, fully left bounded left Noetherian rings, especially finitely generated Noetherian algebras, are considered. It is noted that for such rings every idempotent kernel functor a is symmetric, i.e. the filter T(σ) of a-dense left ideals has a basis of a-dense ideals. A prime ideal P of a f.l.b.l.N. ring R is localizable if and only if it is the intersection of the P-critical left ideals. In case R is a finitely generated algebra over its (Noetherian) center C, we apply the technique of “descent” of kernel functors. If a is a symmetric kernel functor such that R(A n c) S T(σ) for every A G T(σ) and such that a has property (T) then there is a kernel functor a’ on C-modules such that Qσ (R) ≅Q≅ ,(R). If P is a prime ideal of R then σ- descends to C if and only if P is localizable. Secondly, a class of rings is described in terms of the Zariski topology on Spec. The imposed condition is weaker than maxi...
TL;DR: In this paper, the Specht modules and the radical of the group ring over the symmetric group γp have been studied in the context of symmetric groups and groups.
Abstract: (1976). Specht modules and the radical of the group ring over the symmetric group γp. Communications in Algebra: Vol. 4, No. 5, pp. 435-457.