Journal of Algebra and Its Applications ( IF 0.8 ) Pub Date : 2021-05-25 , DOI: 10.1142/s0219498822501614 M. Zafrullah 1
Let be an integral domain with quotient field throughout Call two elements -coprime if Call a nonzero non-unit of an integral domain rigid if for all we have or Also, call semirigid if every nonzero non-unit of is expressible as a finite product of rigid elements. We show that a semirigid domain is a GCD domain if and only if satisfies product of every pair of non--coprime rigid elements is again rigid. Next, call a valuation element if for some valuation ring with and call a VFD if every nonzero non-unit of is a finite product of valuation elements. It turns out that a valuation element is what we call a packed element: a rigid element all of whose powers are rigid and is a prime ideal. Calling a semi-packed domain (SPD) if every nonzero non-unit of is a finite product of packed elements, we study SPDs and explore situations in which a variant of an SPD is a semirigid GCD domain.
中文翻译:
半刚性 GCD 结构域 II
让是具有商场的整数域始终调用两个元素-互质如果调用非零非单元积分域的如果对所有人都僵硬我们有或者另外,打电话半刚性,如果每个非零非单位可表示为刚性单元的有限乘积。我们证明了一个半刚性域是一个 GCD 域当且仅当满足每对非的乘积-coprim 刚性元素又是刚性的。接下来,调用一个估值元素如果对于一些估价戒指和并打电话一个 VFD,如果每个非零非单位是估值元素的有限乘积。事实证明,估值元素就是我们所说的打包元素:刚性元素他们所有的权力都是刚性的是一个主要理想。打电话如果每个非零非单元是填充元素的有限乘积,我们研究 SPD 并探索 SPD 的变体是半刚性 GCD 域的情况。