Ore扩张环的弱相伴素理想

Weak Associated Primes over Ore Extension Rings

作为对零化子与相伴素理想概念的推广,该文引进了弱零化子与弱相伴素理想的定义,并探讨了环R的弱相伴素理想与它的Ore扩张环R[x;α,δ]的弱相伴素理想之间的关系.证明了如果环R是(α,δ)-相容的可逆环,那么NAss(R[x;α,δ])={P[x;α,δ]| P∈NAss(R)},其中NAss(R[x;α,δ])与NAss(R)分别是环R[xα,δ]与环R的所有弱相伴素理想集合.这样Ore扩张环R[x;α,δ]的弱相伴素理想就直接可以用环R的弱相伴素理想来刻画.从而将文献[3,5 6]中的相关结论推广到更一般的情形.

As a generalization of annihilator and associated primes, in this paper, the author introduces the notions of weak annihilator and weak associated primes, and investigates the relationship between the weak associated primes of a ring R and those of the Ore extension ring R[x; α,δ]. It is proved that if R is an （α,δ）-compatible reversible ring, then NAss（R[x; α,δ]） = {P[x;α,δ I P e NAss（R）}, where NAss（R[x; α,δ]） and NAss（R） stand for the set of all weak associated primes over R[x; α,δ] and that over R, respectively. So all associ- ated primes over R[x; α,δ] can be described in terms of the weak associated primes over R in a very straightforward way. Consequently, several results in [3], [5], [6] are generalized to a more general setting.