T-Ideals and C-Ideals

Aaron Klein
1979 Proceedings of the Edinburgh Mathematical Society  
1. Given a ring R we consider the category R of R -rings (rings A with given ring-homomorphisms R -* A), and .R-homomorphisms (ring-homomorphisms that form commutative triangles with the given maps from R). All rings are associative and have 1, all homomorphisms send 1 to 1. Wedefineac-i?-ringasanobjectA ini? with a family of maps {p x Ghom/j(A, A)\x G A}. Equivalently, a c-i?-ring is an R-ring A with a binary operation a for all a, a', bE.A, rE.R. Here f denotes the image of r under R-*A and
more » ... r under R-*A and the ring multiplication on A is denoted by juxtaposition. We call the third operation R -composition and we denote by cR the category whose objects are the c-R -rings and whose maps are those maps of R which preserve R -composition. Let A be a c-R -ring and KdA. We call K a c-ideal in A if K is the kernel of a map in cR. The following is implied. Theorem 1. KG A is a c-ideai in A if and only if: Oi) K is an ideal in the ring A, (1 2 ) k • a G K for all k G K, a 6 A, and (1 3 ) a(k + a')-aa'EKfor all a,a'EA,k£ K. 2. We emphasise the importance of c-ideals by relating them to the well-known T-ideals (1,2.2) in free algebras. Let A be a commutative ring with 1 and V = A{Xj} sSS the free associative A-algebra with 1 over a set S (the notation follows (1)). Let V |s| be the direct product of \S\ copies of V. We may write elements of V |s| as vectors of polynomials in the (non-commuting) indeterminates {x s \ s G S}, namely f = (Js(*))ses with x = (x s ) ses , with component-wise addition and multiplication. We define a composition on V |s|
doi:10.1017/s0013091500016199 fatcat:3vb7rbf565firiaix7kiu7udte