On Dedekindian $l$-semigroups and its lattice-ideals
Kentaro Murata · Proceedings of the Japan Academy Series A Mathematical Sciences · 1971
Our main purpose of the present note is to study some lattice-ideals of Dedekindian/-semigroups.The notation and terminology are those of [1].I. Let S be an Artinian -semigroup considered in [1].An inte- gral element q of S is called primary if the conditions xy_ q, x q (x, y la) imply y" q for some positive integer p. Then it can be proved that p sup {x I x" _ q for some positive integer p} is a prime element in I. Now let -{v} be system of valuations with the properties (A), (B)and (C)in [I].Then for any fixed v and for primary element q with q_p(v) (cf. [14]), we have that (q)4=0 nd v'(q)--0 for every ' with '4: v.By using the above fact nd the results in [1; 4], we cn prove that, if p is a low prime element of I, the set of the minimal primes less thn p consists of infinite many members.