Lifting modules plays important roles in module theory. H-supplemented are a nice generalization of lifting which have been studied extensively recently. In this article, we introduce proper via images fully invariant submodules. Let F be submodule right Rmodule M. We say that M is IF -H-supplemented case for every endomorphism φ M, there direct summand D such φ(F) + X = if and only It proved d...

After the paper [1] was published, following information about D-extending modules pointed out by Adnan Tercan. Let $r$ be a left exact preradical in category of right $R$-modules and let $M$ $R$-module. A submodule $N$ is called provided $r(M/N)=0$. module an ES-module every direct summand $M$. ES-modules introduced A. Tercan [2]. Since socle preradical, coincide with for $r=Soc$.The author wh...

We investigate Γ-cohomology of some commutative cooperation algebras E∗E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Γ-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E∞ structu...

Let G be a finite group, and let Ω := {t ∈ G | t = 1}. Then Ω is a G-set under conjugation. Let k be an algebraically closed field of characteristic 2. It is shown that each projective indecomposable summand of the G-permutation module kΩ is irreducible and self-dual, whence it belongs to a real 2-block of defect zero. This, together with the fact that each irreducible kG-module that belongs to...

The purpose of this note is to describe when a general complex algebraic $^*$-algebra pre-$C^*$-normed, and investigate their structure the $^*$-algebras are Baer $^*$-rings in addition algebraicity. As main result we prove following theorem for $^*$-algebras: every kind can be decomposed as direct sum $M\oplus B$, where $M$ finite dimensional $B$ commutative $^*$-algebra. summand $^*$-isomorph...

We prove that if $R$ is a commutative Noetherian ring, then every countably generated flat $R$-module quite flat, i.e., direct summand of transfinite extension localizations in countable multiplicative subsets. also show the spectrum cardinality less than $\kappa$, where $\kappa$ an uncountable regular cardinal, modules with generators. This provides alternative proof fact over ring spectrum, a...

We refine our earlier work on the existence and uniqueness of E∞ structures on K-theoretic spectra to show that at each prime p, the connective Adams summand ℓ has an essentially unique structure as a commutative S-algebra. For the p-completion ℓp we show that the McClure-Staffeldt model for ℓp is equivalent as an E∞ ring spectrum to the connective cover of the periodic Adams summand Lp. We est...

In this paper we study the modular structure of the permutation module H ) of the symmetric group S2n acting on set partitions of a set of size 2n into n sets each of size 2, defined over a field of odd characteristic p. In particular we characterize the vertices of the indecomposable summands of H ) and fully describe all of its indecomposable summands that lie in blocks of p-weight at most tw...