GCD-摹群条件下的半群环主理想升链条件

On Ascending Chain Condition on Principal Ideals of Semigroup Rings When Semigroups Are GCD-Monoids

对交换半群环的主理想升链条件进行了讨论.在环是整环,半群S是交换无挠可消摹群,S中存在完全不可逆生成集的条件下得到一个关于半群环的主理想升链条件的充要条件.在交换摹群S是唯一分解的条件下证明了在S中存在完全不可逆生成集,由此得到交换无挠可消摹群是GCD-摹群的条件下关于半群环的主理想升链条件的充要条件.

The ascending chain condition on principal ideals（a, c. c. p. ） of semigroup rings is discussed. Under the semigroups are commutative torsion-free and cancellative monoids, a necessary and sufficient condition of a. c. c. p. of semigroup rings is obtained when there are complete non-reversible generating sets in the semigroups. And another necessary and sufficient condition of a, c. c.p. of semigroup rings is obtained under the condition that the rings are integral domains and commutative torsion-free and cancellative monoids are GCD- monoids.