Homomorphic Image and Inverse Image of Weak Closure Operations on Ideals of BCK-Algebras

We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras φ : X → Y , we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping c l Y : I ( Y ) → I ( Y ) , we define a map c l Y ← on I ( X ) by A ↦ φ − 1 ( φ ( A ) c l Y ) . We prove that, if “ c l Y ” is a weak closure operation (respectively, semi-prime and meet) on I ( Y ) , then so is “ c l Y ← ” on I ( X ) . In addition, for mapping c l X : I ( X ) → I ( X ) , we define a map c l X → on I ( Y ) as follows: B ↦ φ ( φ − 1 ( B ) c l X ) . We show that, if “ c l X ” is a weak closure operation (respectively, semi-prime and meet) on I ( X ) , then so is “ c l X → ” on I ( Y ) .

Some remarks on subgroups of maximal class in a finite p-group

The following characterizations of p-groups of maximal class are proved: (a) If a p-group of order > pp+2 contains a subgroup of maximal class and index p, then G possesses at most one normal subgroup of order pp and exponent p. (b) If the center of any nonabelian epimorphic image of a nonabelian two-generator p-group G is cyclic, then either G ≅ M pn or G is of maximal class. (c) An 𝒜n-group G, n > 1, is of maximal class ⇔ all its 𝒜2-subgroups of minimal order are of maximal class. (iv) If all factors of the lower central series of a nonabelian two-generator p-group are cyclic, then it is either of maximal class or ≅ M pn. (v) If a nonabelian p-group G is such that any s pairwise non-commuting elements generate a group of maximal class, where s is the fixed member of the set {3, …, p + 1} and p > 2 if s ≠ p + 1, then G is also of maximal class. We also study the noncyclic p-groups containing only one normal subgroup of a given order.

CATEGORIES OF COMODULES AND CHAIN COMPLEXES OF MODULES

Let [Formula: see text] denote the coendomorphism left R-bialgebroid associated to a left finitely generated and projective extension of rings R → A with identities. We show that the category of left comodules over an epimorphic image of [Formula: see text] is equivalent to the category of chain complexes of left R-modules. This equivalence is monoidal whenever R is commutative and A is an R-algebra. This is a generalization, using entirely new tools, of results by Pareigis and Tambara for chain complexes of vector spaces over fields. Our approach relies heavily on the noncommutative theory of Tannaka reconstruction, and the generalized faithfully flat descent for small additive categories, or rings with enough orthogonal idempotents.

ON KRULL–SCHMIDT FINITELY ACCESSIBLE CATEGORIES

Let 𝒞 be a finitely accessible additive category with products, and let (Ui)i∈Ibe a family of representative classes of finitely presented objects in 𝒞 such that each objectUiis pure-injective. We show that 𝒞 is a Krull–Schmidt category if and only if every pure epimorphic image of the objectsUiis pure-injective.

From Farey symbols to generators for subgroups of finite index in integral group rings of finite groups

The topic of this paper is the construction of a finite set of generators for a subgroup of finite index in the unit group u(ℤG) of the integral group ring of a finite group G. The present paper is a continuation of earlier research by Bass and Milnor, Jespers and Leal, and Ritter and Sehgal who constructed such generators provided that the group G does not have a non-abelian fixed-point free epimorphic image and the rational group algebra ℚG does not have simple epimorphic images that are two-by-two matrices over either the rationals, a quadratic imaginary extension of the rationals or a non-commutative division algebra. In this paper we allow simple images of the type M2(ℚ). We will do so by introducing new additional generators using Farey symbols, which are in one to one correspondence with fundamental polygons of congruence subgroups of PSL2(ℤ). Furthermore, for each simple Wedderburn component M2(ℚ) of ℚG, the new generators give a free subgroup that is embedded in M2(ℤ).

A note on the fixed subring of an FPF ring

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

Groups of height four

If G and H are infinite groups then G is said to be larger than H (H≼G) if there are subgroups A of G, B of H, each of finite index, such that B is an epimorphic image of A. Pride (1979) showed that if G has finite ‘height’ with respect to the quasi-order ≼ then there are only finitely many (classes of) minimal groups H with H ≼G, and asked whether this were true without the minimality restriction on H. This paper gives a negative answer to his question by exhibiting a group G of height four with infinitely many (classes of) groups H satisfying H≼G.1980 Mathematics subject classification (Amer. Math. Soc.): 20 E 99, 20 K 15.

Linear monads

A monad T = (T, μ, η) on a category C is said to be linear with respect to a dense functor N: A → C if the operator T is the epimorphic image of a certain colimit of its values on A. The main aim of the article is to relate the concept of a linear monad to that of a monad with rank. A comparison is then made between linear monads and algebraic theories.

A Note on Group Rings of Certain Torsion-Free Groups

As a step towards characterizing ID-groups (i.e., groups G such that, for every ring R without zero-divisors, the group ring RG has no zero-divisors), Rudin and Schneider defined Ω-groups, a possibly wider class than that of right-orderable groups, and proved that if every non-trivial finitely generated subgroup of a group G has a non-trivial H-group as an epimorphic image, then G is an ID-group. We prove that such groups are even Ω-groups and obtain the analogous result for right-orderable groups.