Algebra Colloquium

A generalization of Witt's theorem is given. The equivalence of the generalization of Witt's theorem over ℝ and Sylvester's law of nullity is proved.

Let A and B be subgroups of a group G, and ∅ ≠ X ⊆ G. Then A is said to be X-permutable with B if there exists an element x ∈ X such that AB^x = B^xA. In this paper, a new version of Schur–Zassenhaus theorem is obtained for X-permutable subgroups, and hence, Question 5.1 in [6] is answered.

A ring R is said to be a generalized stable ring provided that aR + bR = R with a, b ∈ R implies that there exists y ∈ R such that a + by ∈ K(R), where K(R) = {x ∈ R | ∃s, t ∈ R such thatsxt = 1}. Let A be a quasi-projective right R-module, and let E = End_R(A). If E is an exchange ring, then E is a generalized stable ring if and only if for any R-morphism f : A → M with Im f ≤^⊕M and any R-epimorphism g : A → M, there exist e = e² ∈ E and h ∈ K(E) such that f = g(eh). Furthermore, we prove that every regular matrix over a generalized stable exchange ring admits a diagonal reduction by quasi-invertible matrices.

Let S be a semigroup. The degree of S is the smallest natural number r such that for each x ∈ S, x^n(x)+r = x^n(x), where n(x) ∈ ℕ. If such a number r does not exist, we say that the degree of S is infinite. For a group G, this coincides with the exponent of G. We prove that for a periodic ring R, the degree of R equals exp(U(R)), where U(R) denotes the unit group of R. Then we determine all degrees for any rings.

It is proved that the Yoneda Ext-algebras of Morita equivalent semiperfect algebras are graded equivalent. The Yoneda Ext-algebras of noetherian semiperfect algebras are studied in detail. Let A be a noetherian semiperfect algebra with Jacobson radical J. We construct a right ideal of the Yoneda algebra , which plays an important role in the discussion of the structure of E(A). An extra grading is introduced to , by which we give a description of the right ideal of E(A) generated by , and we give a necessary and sufficient condition for a notherian semiperfect algebra to be higher quasi-Koszul. Finally, it is shown that the quasi-Koszulity of a noetherian semiperfect algebra is a Morita invariant.

The superposition of cooperations satisfies the well-known clone axioms (C1), (C2) and (C3). We define terms for indexed coalgebras of type τ, cooperations induced by those terms, and prove that the set of all induced cooperations forms a clone. This clone is equal to the clone of all cooperations generated by the fundamental cooperations of an indexed coalgebra. Finally, we introduce the concept of rational equivalence for coalgebras and determine all two-element coalgebras up to rational equivalence.

Let be the Lie algebra with basis {L_i,j, C|i, j ∈ ℤ} and relations [L_i,j, L_k,l] = ((j + 1)k - i(l + 1))L_{i+k, j+l} + iδ_{i, -k}δ_{j+l, -2}C and [C, L_i,j] = 0. It is proved that an irreducible highest weight -module is quasifinite if and only if it is a proper quotient of a Verma module. An additive subgroup Γ of 𝔽 corresponds to a Lie algebra of Block type. Given a total order ≻ on Γ and a weight Λ, a Verma -module M(Λ, ≻) is defined. The irreducibility of M(Λ, ≻) is completely determined.

In this article we give a self-contained existence and uniqueness proof for the Tits simple group T. Parrott gave the first uniqueness proof. Whereas Tits' and Parrott's results employ the theory of finite groups of Lie type, our existence and uniqueness proof follows from the general algorithms and uniqueness criteria for abstract finite simple groups described in the first author's book [11]. All we need from the previous papers is the fact that the centralizer H of the Tits group T is an extension of a 2-group J with order 2⁹ and nilpotency class 3 by a Frobenius group F of order 20 such that the center Z(H) has order 2 and any Sylow 5-subgroup Q of H has a centralizer C_J(Q) ≤ Z(H).

An algebraic deformation theory of dialgebra morphisms is obtained.

In this paper, we extend the study of various annihilator conditions on the nearring of polynomials and the nearring of formal power series to skew polynomials and skew formal power series in which addition and substitution are used as operations. A result of Hong et al. on the skew polynomial rings and skew power series of an α-rigid Baer ring is extended to Baer conditions in the nearrings of skew polynomials and skew formal power series. Also, for an injective endomorphism α of a ring R, it is shown that R is α-rigid if and only if the nearring (R[x; α], +, ◦) is reduced, if and only if the nearring (R₀[[x; α]], +, ◦) is reduced, if and only if R is a reduced and near Armendariz ring.

The study of the cohomological dimension of algebraic varieties has produced some interesting results and problems in local algebra. Let 𝔞 be an ideal of a commutative Noetherian ring R. For finitely generated R-modules M and N, the concept of cohomological dimension cd_𝔞(M, N) of M and N with respect to 𝔞 is introduced. If 0 → N' → N → N'' → 0 is an exact sequence of finitely generated R-modules, then it is shown that cd_𝔞(M, N) = max{cd_𝔞(M, N'), cd_𝔞(M, N'')} whenever proj dim M < ∞. Also, if L is a finitely generated R-module with Supp(N/Γ_𝔞(N)) ⊆ Supp(L/Γ_𝔞(L)), then it is proved that cd_𝔞(M, N) ≤ max{cd_𝔞(M,L), proj dim M}. Finally, as a consequence, a result of Brodmann is improved.

In this paper, we study the representations of a class of non-commutative associative algebras A_Q related to the quantum torus , which is introduced by Berman, Gao and Krylyuk to study the extended affine Lie algebras. The representations of A_Q are important for the study of the representations of certain “half lattice” vertex algebras.

In this paper, we prove that for any prime p, a finite group with 18p elements having maximal order is solvable. This result provides some information in solving an open problem proposed by Thompson.

We give a short proof for the associativity of the binary operation defined on a refined system of semigroups indexed by a semilattice. The main result given by Zhang, Shum and Zhang in 2001 on the refined semilattice of semigroups is substantially strengthened.

The class of groups having a series with finite factors is not quotient closed. In this note, we prove that if the group G has a series whose factors are finite and H is a locally nilpotent normal subgroup of G, then G/H also has a series whose factors are finite.

Let (R, 𝔪) be a complete Noetherian local ring and let M be a finite R-module of positive Krull dimension n. It is shown that any subset T of Assh_R(M) can be expressed as the set of attached primes of the top local cohomology module for some ideal 𝔞 of R. Moreover, if 𝔞 is an ideal of R such that the set of attached primes of is a non-empty proper subset of Assh_R(M), then for some ideal 𝔟 of R with dim_R(R/𝔟) = 1.

This note presents a counterexample to [1, Proposition 1(d)] that was communicated by Külshammer. The proof of [1, Proposition 1(d)] is erroneous.

The classification of nilpotent Lie algebras of maximal rank was solved by Santharoubane. In the present paper, we prove that the classification of non-degenerate solvable Lie algebras of maximal rank can be obtained from the work of Santharoubane.