# Algebra II. Ring Theory: Ring Theory by Carl Faith By Carl Faith

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Sample text

EB {~}. The comonolith is therefore the only coatom in Con L, when such exists. 14 If L E Kw then the follOWing statements are equivalent: (1) K(L) == T 2 (L); (2) Con L has comonolith (2) : Since (0,1) ¢

Km,1l and is DOl-generated by {fj(Xj)IO~j<2m+n, 1~i~k}. The result now follows from the fact that DOl is locally finite. <> Our objective now is to show that for every algebra in Kw (and therefore every algebra that belongs to a Berman class) the properties of being finitely subdirectly irreducible and subdirectly irreducible are equivalent. 8 If an Ockham algebra L is finitely subdirectly irreducible then every CPI -class in L contains at most two elements. Proof Suppose that a CPt -class contains at least three elements.

Clearly, a II (3(P) and b II (3(P) belong to B, and (a II (3(P), b II (3(P») E fJ(a, b). c ~ If a II (3(P) < b II (3(P), let c be an atom of B with c 1; a II (3(P) and b II (3(P). (a), c) = (a II (3(P) II c, b II (3(P) II c) E fJ(a, b). (a), (3(c» E 'I9(a, b). (a) < (3(c), we deduce that (0,1) E 'I9(a, b) and that therefore 1J(a, b) = t. If now a II (3(P) = b II (3(P) let a 1 = a V (3(P) < b V (3(P) = b l . Then (al, hd E '19 (a , b). Moreover, we cannot have (3(P) = 1, so (3(P) must be a fixed point.