零化子乘法模：环的刻画与构造

中文摘要 AI 辅助

一个 $A$-模 $E$ 称为零化子乘法模，如果每个元素的零化子等于某个有限生成理想 $I$ 的 $IE$ 的零化子。我们刻画了每个忠实模都具有此性质的环，即交换拟 Frobenius 环；以及每个模都具有此性质的环，即 Artin 主理想环。这两个刻画都归结为二生成模的情形。对于具有有限多个极小素理想的约化环，忠实零化子乘法模恰好是正则无挠模，且定义中只需主理想即可。零化子理想上的升链条件给出了模零化子的有限检测，从而得到局部化和图刚性结果。我们还证明了结合素理想与忠实商模的结合素理想相等，并建立了射影张量和迹的刻画。在局部平方零环上，该性质等价于一个双线性乘法映射的非奇异性。一个射影维数论证给出了一个精确的长度界，该界由显式的忠实不可分解非射影模达到，并计算了其自同态环。这些例子的每个循环子模都嵌入环中，尽管整个模不是无挠的。支撑、遗传挠理论和并合准则将这些结构结果与进一步的模构造联系起来。

英文摘要

An $A$-module $E$ is annihilator multiplication if the annihilator of each element equals that of $IE$ for a finitely generated ideal $I$. We characterize rings for which every faithful module has this property as the commutative quasi-Frobenius rings, and rings for which every module has the property as the Artinian principal ideal rings. Both characterizations reduce to two-generated modules. Over a reduced ring with finitely many minimal primes, faithful annihilator multiplication modules are exactly the regular-torsion-free modules, and principal ideals suffice in the definition. An ascending chain condition on annihilator ideals gives finite detection of the module annihilator, yielding localization and graph rigidity results. We also prove equality of associated primes with those of the faithful quotient and establish projective tensor and trace characterizations. Over local square-zero rings, the property is equivalent to nonsingularity of a bilinear multiplication map. A projective dimension argument gives a sharp length bound, attained by explicit faithful indecomposable nonprojective modules whose endomorphism rings are computed. Every cyclic submodule of these examples embeds in the ring, although the whole module is not torsionless. Support, hereditary torsion, and amalgamation criteria connect these structural results with further module constructions.

发表机构

• Hoseo University（湖西大学）
• Marmara University（马尔马拉大学）

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