Incidence rings and their automorphisms

The paper is devoted to automorphisms of incidence algebras. Let I(X, R) be an incidence algebra of a preordered set X over a T-algebra R (T is a commutative ring). The algebra I(X, R) is assumed to satisfy a condition of sufficiently general character. It is called condition (II). In the case when the algebra I(X, R) satisfies condition (II), it is proved that any automorphism of such algebra after conjugation by an inner automorphism has a diagonalized form in a certain sense (Theorem 3.1). The other two main results of the paper are Theorem 4.1 and Corollary 4.2. In these propositions, in addition to condition (II), the algebra I(X, R) satisfies two other certain conditions. All three conditions are fulfilled if, for example, R is a local ring or a domain of principal left (right) ideals. Under these assumptions, it is proved that every automorphism of the algebra I(X, R) can be written as a product of inner, multiplicative, ring, and order automorphisms. These four kinds of automorphisms can be called standard. Here we consider an automorphism to be standard, if its structure is quite clear.

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