By Haskell Curry

ISBN-10: 0268002746

ISBN-13: 9780268002749

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But ek = (e . e) k = e(ek) = ere and therefore ek E eRe. To show that #J is onto, let ere E eRe. Then the mapping t,k ae-ae * ere (7) for a E R is an element of HomR(re,Re) since by (7) x(ae) = (xu) e --+ xae . ere = x(ae * ere) The image of e under 4 is ere showing that #J is onto. We have still to show that #J is a homomorphism. Let ki E K and eki = er,e. Then #J(kl + k,) = e(kl + k,) + = ek, + ek, = erle + erze = 4kl #Jkz #J(klk,) = (ek1) k, = @le> k2 = (er,e)(ek,) = (er1e>(erze)= (#Jk1)(4kz) 1 50 11.

If H = M , an R-module, there is a one-one correspondence between the congruence relations p on M and the R-submodules of M ; for let p be a congruence relation on M and let the kernel ker p of this relation be the pre-image of the 0 element of M under the natural epimorphism of H to H l p associated with p. Then ker(pl n pz) = ker p1 n ker pz and ker(pl u pz) = ker p1 ker p2 . The analogous result holds for congruence relations on rings. In this case ker p is an ideal. It is to be noted, however, that not all congruence relations on a semigroup are uniquely determined by their kernels.

X y E U(ab)]. , u,) is an arbitrary finite set of elements of M. , u,} coincides with that of a. Condition Ul holds trivially for these families of subsets. , u,) be two basic neighborhoods of p . , v,) _C U n V. , u,) E 3;(q). , n and hence x E U(p). The continuity of subtraction relative to this topological structure follows immediately. , u,). , u,). To prove the continuity of multiplication we note first that the product ab of two elements in Hom,(M, M ) is defined by m -ta(bm). Therefore, 30 11.

### A theory of formal deducibility by Haskell Curry

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