Star Projective and Star Injective Hv-Modules

In this paper, we introduce the concepts of product and directsum, star projective and star injective in $H_v$-modules. Weinvestigate generalizations of some notions in homological algebrato prove the five lemma and star projective and star injectivetheorems in $H_v$-modules. We determine equivalent conditions for split sequences in $H_v$-modules and present some related results.

PERUMUMAN LEMMA SNAKE DAN LEMMA LIMA

Abstrak. Barisan -eksak merupakan perumuman dari barisan eksak yang diperkenalkan oleh Davvaz dan Parnian-Garamaleky. Dalam tulisan ini akan dikaji perumuman dari Lemma Snake dan Lemma Lima yang memanfaatkan sifat-sifat dari barisan -eksak. Kata Kunci : Barisan -eksak, Lemma Snake, Lemma Lima Abstract. -exact sequences was introduced by Davvaz dan Parnian-Garamaleky as a generalization of exact sequences. In this paper, we give some characterizations and properties of -exact sequences. We use these result to find a generalization of Snake Lemma and Five Lemma. Key words Kunci : -exact sequences, Snake Lemma, Five Lemma

A five lemma for free products of groups with amalgamations

This paper explores a five-lemma situation in the context of a free product of a family of groups with amalgamated subgroups (that is, a colimit of an appropriate diagram in the category of groups). In particular, for two families {Aα}, {Bα} of groups with amalgamated subgroups {Aαβ}, {Bαβ} and free products A, B we assume the existence of homomorphisms Aα → Bα whose restrictions Aαβ → Bαβ are isomorphisms and which induce an isomorphism A → B between the products. We show that the usual five-lemma conclusion is false, in that the morphisms Aα → Bα are in general neither monic nor epic. However, if all Bα → B are monic, Aα → Bα is always epic; and if Aα → A is monic, for all α, then Aα → Bα is an isomorphism.