Showing papers on "Idempotence published in 1976"

Subalgebras that are cyclic as submodules

Abstract: Let R be an associative, commutative, unital ring. By a R-algebra we mean a unital R-module A together with a R-module homomorphism μ: ⊗ R n A→A (n≥2). We raise the question whether such an algebra possesses either an idempotent or a nilpotent element. In section 1 an affirmative answer is obtained in case R=k is an algebraically closed field and dimkA<∞, as well as in case R=ℝ, dimℝS<∞, and n≡0(2). Section 2 deals with the case of reduced rings R and R-algebras which are finitely generated and projective as R-modules. In section 3 we show that the “generic” algebra over an integral domain D fails to have nilpotent elements in any integral domain extending its base ring Dn,m, and thus acquires an idempotent element in some integral domain extending Dn,m.

Inertial coefficient rings and the idempotent lifting property

Abstract: A commutative ring R with identity is called an inertia! coefficient ring if every finitely generated ^-algebra A with A/N separable over R contains a separable ^-subalgebra S of A such that A = S + N, where N is the Jacobson radical of A. We say A has the idempotent lifting property if every idempotent in A/N is the image of an idempotent in A. Our main theorem is that any finitely generated algebra over an inertial coefficient ring has the idempotent lifting property. All rings contain an identity; all subrings contain the identity of the overring; all homomorphisms preserve the identity. Throughout R denotes a commutative ring and A an R -algebra which is finitely generated as an fi-module. The Jacobson radical of a ring B is denoted rad(fi) and throughout rad(A) = A. A separable fi-subalgebra 5 of A such that A = S + A is called an inertial subalgebra. If every finitely generated fi-algebra A with A/N fi-separable has an inertial subalgebra, R is called an inertial coefficient ring. The basic properties of inertial subalgebras and inertial coefficient rings can be found in [7]. If / is an ideal of a ring B we call (B, I) an L. I. pair (lifting idempotent pair) if every idempotent in the factor ring fi// is the image of an idempotent in B; if (A, A) is an L. I. pair we say A has the idempotent lifting property. Our main theorem is motivated by a conjecture of E. C. Ingraham that if every finitely generated fi-algebra has the idempotent lifting property then R is an inertial coefficient ring. Our proof of the converse of this conjecture has as a corollary that an inertial coefficient ring is a Hensel ring (see [5], [6], and [10] for definition and properties of Hensel rings), recalling the role Hensel local rings have played in generalizations of the Wedderburn Principal Theorem by Azumaya, Ingraham, and W. C. Brown. A second immediate consequence of our main theorem is that when R is an inertial coefficient ring, two inertial subalgebras of an fi-algebra A are conjugate under an inner automorphism of A, generalizing Malcev's uniqueness statement to Wedderburn's Principal Theorem. Received by the editors April 14, 1976. AMS (MOS) subject classifications (1970). Primary 16A16, 16A32; Secondary 13J15.

Note on the orderability of idempotent semigroups

Abstract: In [3] , we gave a condition for the orderability of finite idempotent semigroups. Recently, in [5], T. Saito gives a condition for the orderability of idempotent semigroups in the general case. As Corollary ( 4–13 of [5] ), he obtains an idempotent semigroup S is orderable if and only if every finite subsemigroup of S is orderable. The purpose of this note is to give a direct proof of this result.