164 research outputs found
Regular pairings of functors and weak (co)monads
For functors L : A → B and R : B → A between any categories A and B, a pairing is defined by maps, natural in A ∈ A and B ∈ B,
MorB(L(A), B) ↔ MorA(A, R(B)).
(L, R) is an adjoint pair provided α (or β) is a bijection. In this case the composition RL defines a monad on the category A, LR defines a comonad on the category B, and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors.
For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of weak monads and weak comonads and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors (L, R, α, β) with α = α ⋅ β ⋅ α and β = β ⋅ α ⋅ β. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on A gives rise to a regular pairing between A and the category of compatible (co)modules
Correct classes of modules
For a ring R, call a class C of R-modules (pure-)
mono-correct if for any M, N ∈ C the existence of (pure) monomorphisms M → N and N → M implies M ≃ N. Extending results
and ideas of Rososhek from rings to modules, it is shown that, for
an R-module M, the class σ[M] of all M-subgenerated modules
is mono-correct if and only if M is semisimple, and the class of
all weakly M-injective modules is mono-correct if and only if M is
locally noetherian. Applying this to the functor ring of R-Mod provides a new proof that R is left pure semisimple if and only if R-Mod
is pure-mono-correct. Furthermore, the class of pure-injective Rmodules is always pure-mono-correct, and it is mono-correct if and
only if R is von Neumann regular. The dual notion epi-correctness
is also considered and it is shown that a ring R is left perfect if
and only if the class of all flat R-modules is epi-correct. At the end
some open problems are stated
Weak Frobenius monads and Frobenius bimodules
As observed by Eilenberg and Moore (1965), for a monad F with right adjoint comonad G on any category A, the category of unital F-modules AF is isomorphic to the category of counital G-comodules AG. The monad F is Frobenius provided we have F=G and then AF≃AF. Here we investigate which kind of isomorphisms can be obtained for non-unital monads and non-counital comonads. For this we observe that the mentioned isomorphism is in fact an isomorphisms between AF and the category of bimodules AFF subject to certain compatibility conditions (Frobenius bimodules). Eventually we obtain that for a weak monad (F,m,η) and a weak comonad (F,δ,ε) satisfying Fm⋅δF=δ⋅m=mF⋅Fδ and m⋅Fη=Fε⋅δ, the category of compatible F-modules is isomorphic to the category of compatible Frobenius bimodules and the category of compatible F-comodules
Irreducible actions and compressible modules
Any finite set of linear operators on an algebra yields an operator
algebra and a module structure on A, whose endomorphism ring is isomorphic
to a subring of certain invariant elements of . We show that if is
a critically compressible left -module, then the dimension of its
self-injective hull over the ring of fractions of is bounded by the
uniform dimension of and the number of linear operators generating .
This extends a known result on irreducible Hopf actions and applies in
particular to weak Hopf action. Furthermore we prove necessary and sufficient
conditions for an algebra A to be critically compressible in the case of group
actions, group gradings and Lie actions
Characterizing rings in terms of the extent of injectivity and projectivity of their modules
Given a ring R, we define its right i-profile (resp. right p-profile) to be
the collection of injectivity domains (resp. projectivity domains) of its right
R-modules. We study the lattice theoretic properties of these profiles and
consider ways in which properties of the profiles may determine the structure
of rings and viceversa. We show that the i-profile is isomorphic to an interval
of the lattice of linear filters of right ideals of R, and is therefore modular
and coatomic. In particular, we give a practical characterization of the
i-profile of a right artinian ring. We show through an example that the
p-profile is not necessarily a set, and also characterize the right p-profile
of a right perfect ring. The study of rings in terms of their (i- or p-)profile
was inspired by the study of rings with no (i- or p-) middle class, initiated
in recent papers by Er, L\'opez-Permouth and S\"okmez, and by Holston,
L\'opez-Permouth and Orhan-Ertas. In this paper, we obtain further results
about these rings and we also use our results to provide a characterization of
a special class of QF-rings in which the injectivity and projectivity domains
of any module coincide.Comment: 19 pages, examples and propositions added. Title change
τ-complemented and τ-supplemented modules
Proper classes of monomorphisms and short exact
sequences were introduced by Buchsbaum to study relative homological algebra. It was observed in abelian group theory that complement submodules induce a proper class of monomorphisms and
this observations were extended to modules by Stenstr¨om, Generalov, and others. In this note we consider complements and supplements with respect to (idempotent) radicals and study the related
proper classes of short exact sequences
Pre-torsors and Galois comodules over mixed distributive laws
We study comodule functors for comonads arising from mixed distributive laws.
Their Galois property is reformulated in terms of a (so-called) regular arrow
in Street's bicategory of comonads. Between categories possessing equalizers,
we introduce the notion of a regular adjunction. An equivalence is proven
between the category of pre-torsors over two regular adjunctions
and on one hand, and the category of regular comonad arrows
from some equalizer preserving comonad to on
the other. This generalizes a known relationship between pre-torsors over equal
commutative rings and Galois objects of coalgebras.Developing a bi-Galois
theory of comonads, we show that a pre-torsor over regular adjunctions
determines also a second (equalizer preserving) comonad and a
co-regular comonad arrow from to , such that the
comodule categories of and are equivalent.Comment: 34 pages LaTeX file. v2: a few typos correcte
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