多项式与幂级数扩张中的（局部）相伴子环
(Locally) Associated Subrings in Polynomial and Power Series Extensions
AI总结：
本文针对交换代数中用于构造反例的广义多项式与幂级数环，建立相关记号与初步结果，给出其在更大同类环中为（局部）相伴的充要条件，助力理解形式幂级数环的半因子性。
AI中文摘要：
子环的相伴性、保理想性与局部相伴性于2024年被正式定义，可通过大环信息理解子环的乘法结构。本文为交换代数领域中常用于构造反例的一类广义多项式环与幂级数环建立记号与初步结果，给出此类多项式环或幂级数环在更大同类环中为（局部）相伴的充要条件，由此可构造多个有价值的例子，并加深对数域中序R上的形式幂级数环R[[x]]何时为半因子环的理解。
英文摘要：
The associated, ideal-preserving, and locally associated properties of subrings, first formally defined in 2024, give a way of understanding the multiplicative structure of a subring given information about the larger ring. In this paper, we establish notation and preliminary results on a generalized type of polynomial and power series rings that is often used in the construction of counterexamples in the field of commutative algebra. We then provide sets of necessary and sufficient conditions under which such a polynomial or power series ring may be (locally) associated in a larger such ring. This allows for the production of several informative examples, as well as a better understanding of the circumstances under which the ring of formal power series $R[[x]]$ over an order $R$ in a number field may be half-factorial.
