Journal of Algebra and Its Applications

Let B be the Burnside ring considered as a globally-defined Mackey functor, and k be a field of positive characteristic q. We prove that, when k ⊗_ℤ B is restricted to nilpotent groups of order prime to q, the only simple Mackey functors that can appear as subquotients of it are of the form S_E,k, where E is the direct product of elementary abelian p-groups. In the special case k = 𝔽₂, the simple Mackey functor S_E,𝔽2 must appear as a filtration factor in 𝔽₂ ⊗_ℤ B for every direct product of elementary abelian p-groups E of odd order.

We introduce the concepts of τ-Max modules and left τ-Max rings which are torsion-theoretic analogues of Max modules and left Max rings. A generalization is obtained of an important theorem by Shock and used to characterize τ-Noetherian rings using τ-Max modules. We then characterize left τ-Max rings and obtain a torsion-theoretic version of a result by Hirano. We conclude with results on τ-short modules, introduced as a torsion analogue of a concept recently defined by Bilhan and Smith.

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is a direct sum of a projective module and a module Q, where Q is either injective or noetherian. This provides an affirmative answer to a question raised by P. F. Smith.

Let p be a prime number and let ℚ/ℤ′ be the elements in ℚ/ℤ of order prime to p. Let , where c is a prime power of p. We use characters and valuation theory to prove that Δ is a parameter space for the cyclic tame extensions of the formal Laurent series field k_p((t)) of degree prime to p. Furthermore, we construct the cyclic tame extension corresponding to a given triple in Δ. The structure of finite cyclic tame extensions of the p-adic number fields was thoroughly investigated by A. A. Albert in 1935. Here we get the same result as consequence of our main theorem.

Lie color algebras are generalizations of Lie superalgebras and graded Lie algebras. The properties of a Lie color algebra can often be related directly to the ring structure of its universal enveloping algebra. We study the effects of torsion elements and torsion subspaces.

Let denote a Lie color algebra. If is homogeneous and torsion then x² = 0 in . If no homogeneous element of is torsion, then so is semiprime. In this case we can give a test which uses Gröbner basis methods to determine when is a domain. This is applied in an example to show may be a domain even if contains torsion elements and torsion subspaces.

Over an arbitrary ring R, a symmetric radical is shown to be strongly normalizing. For a fully semiprimary Noetherian ring R, a symmetric radical is normalizing if and only if the class of torsion factor rings of R is closed under ring isomorphisms. In case S is a strongly normalizing or normalizing extension ring of R, a symmetric radical for S is constructed as an extension of a symmetric radical for R. Applications address questions concerning the behavior of Krull dimension and linked prime ideals of S.

Let (R,𝔪) be a 1-dimensional Cohen–Macaulay local ring of multiplicity e and embedding dimension ν ≥ 2. Let B denote the blowing-up of R along 𝔪 and let I be the conductor of R in B. Let x ∈ 𝔪 be a superficial element in 𝔪 of degree 1 and , . We assume that the length . This class of local rings contains the class of 1-dimensional Gorenstein local rings (see 1.5). In Sec. 1, we prove that (see 1.6) if the associated graded ring G = gr_𝔪(R) is Cohen–Macaulay, then I ⊆ 𝔪^s + xR, where s is the degree of the h-polynomial h_R of R. In Sec. 2, we give necessary and sufficient conditions (see Corollaries 2.4, 2.5, 2.9 and Theorem 2.11) for the Cohen–Macaulayness of G. These conditions are numerical conditions on the h-polynomial h_R, particularly on its coefficients and the degree in comparison with the difference e - ν. In Sec. 3, we give some conditions (see Propositions 3.2, 3.3 and Corollary 3.4) for the Gorensteinness of G. In Sec. 4, we give a characterization (see Proposition 4.3) of numerical semigroup rings which satisfy the condition .

The commuting graph of a group G, denoted by Γ(G), is a simple undirected graph whose vertices are all non-central elements of G and two distinct vertices x, y are adjacent if xy = yx. The commuting graph of a subset of a group is defined similarly. In this paper we investigate the properties of the commuting graph of the symmetric and alternating and subsets of transpositions and involutions in the symmetric groups.