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Phantom maps and purity in modular representation theory, I

D. Benson, G. Gnacadja (1999)

Fundamenta Mathematicae

Let k be a field and G a finite group. By analogy with the theory of phantom maps in topology, a map f : M → ℕ between kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. We investigate the relationships between the theory of phantom maps, the algebraic theory of purity, and Rickard's idempotent modules. In general, adding one to the pure global dimension of kG gives an upper bound for the number of phantoms we need...

P-injective group rings

Liang Shen (2020)

Czechoslovak Mathematical Journal

A ring R is called right P-injective if every homomorphism from a principal right ideal of R to R R can be extended to a homomorphism from R R to R R . Let R be a ring and G a group. Based on a result of Nicholson and Yousif, we prove that the group ring RG is right P-injective if and only if (a) R is right P-injective; (b) G is locally finite; and (c) for any finite subgroup H of G and any principal right ideal I of RH , if f ∈ Hom R ( I R , R R ) , then there exists g ∈ Hom R ( RH R , R R ) such that g | I = f . Similarly, we also obtain equivalent characterizations...

Restricted Boolean group rings

Dinesh Udar, R.K. Sharma, J.B. Srivastava (2017)

Archivum Mathematicum

In this paper we study restricted Boolean rings and group rings. A ring R is 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑𝐵𝑜𝑜𝑙𝑒𝑎𝑛 if every proper homomorphic image of R is boolean. Our main aim is to characterize restricted Boolean group rings. A complete characterization of non-prime restricted Boolean group rings has been obtained. Also in case of prime group rings necessary conditions have been obtained for a group ring to be restricted Boolean. A counterexample is given to show that these conditions are not sufficient.

Rings consisting entirely of certain elements

Huanyin Chen, Marjan Sheibani, Nahid Ashrafi (2018)

Czechoslovak Mathematical Journal

We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ℤ 3 ⊕ ℤ 3 ; ℤ 3 ⊕ B where B is a Boolean ring; local ring with nil Jacobson radical; M 2 ( ℤ 2 ) or M 2 ( ℤ 3 ) ; or the ring of a Morita context with zero pairings where the underlying rings are ℤ 2 or ℤ 3 .

Currently displaying 161 – 180 of 298