In this paper we study the simplicial structure of the complex C • ((A, B, ε); M ), associated to the secondary Hochschild cohomology. The main ingredient is the simplicial object B (A, B, ε), which plays a role equivalent to that of the bar resolution associated to an algebra. We also introduce the secondary cyclic (co)homology and establish some of its properties (Theorems 3.9 and 4.11).

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Abstract. We show that the secondary Hochschild cohomology associated to a triple (A, B, ε) has several of the properties of the usual Hochschild cohomology. Among others, we prove the existence of the cup and Lie products, discuss the connection with extensions of B-algebras, and give a Hodge type decomposition of the secondary Hochschild cohomology.

In this paper we discuss some properties of abelian (weakly) nil clean rings. We prove that any subring of an abelian (weakly) nil clean ring is (weakly) nil clean (Theorem 2). We also show that the tensor product of commutative (weakly) nil clean rings is also (weakly) nil clean and give sufficient conditions for the converse to be true (Theorems 3–6).

A relative derived category for the category of modules over a presheaf of algebras is constructed to identify the relative Yoneda and Hochschild cohomologies with its homomorphism groups. The properties of a functor between this category and the relative derived category of modules over the algebra associated to the presheaf are studied. We obtain a generalization of the Special Cohomology Comparison T heorem of M. Gerstenhaber and S. D.Schack.