论文标题
Semirigid GCD域II
Semirigid GCD domains II
论文作者
论文摘要
令$ d $为商$ k，$ $。 $ d $ semirigid如果每个非零单位的$ d $都是刚性元素的有限产品。我们表明，如果且仅当$ d $满足$ \ ast：$的每对非 - $ V $ - coprime刚性元素的$ $ aste of AST：$的产品再次是刚性的，则仅当$ d $满足$ \ ast：$。下一个呼叫$ a \ in d $ in评估元素，如果$ av \ cap d = ad =某些评估戒指$％v $，$ d \ subseteq v \ subseteq v \ subseteq k $并将$ d $ a vfd拨打，如果$ d $的每个非零单位是估值元素的有限产品。事实证明，评估元素是我们所说的包装元素：一个刚性元素$ r $所有其刚性和$ \ sqrt {rd} $的元素是主要的理想。如果每个非零单位的$ d $是包装元素的有限产品，我们将在半包装域（SPD）致电$ d $，我们研究SPD和探索SPD是半irigid GCD域的情况。
Let $D$ be an integral domain with quotient field $K,$ throughout$.$ Call two elements $x,y\in D\backslash \{0\}$ $v$-coprime if $xD\cap yD=xyD.$ Call a nonzero non unit $r$ of an integral domain $D$ rigid if for all $x,y|r$ we have $x|y$ or $y|x.$ Also call $D$ semirigid if every nonzero non unit of $D$ is expressible as a finite product of rigid elements. We show that a semirigid domain $D$ is a GCD domain if and only if $D$ satisfies $\ast :$ product of every pair of non-$v$-coprime rigid elements is again rigid. Next call $a\in D$ a valuation element if $aV\cap D=aD$ for some valuation ring $% V $ with $D\subseteq V\subseteq K$ and call $D$ a VFD if every nonzero non unit of $D$ is a finite product of valuation elements. It turns out that a valuation element is what we call a packed element: a rigid element $r$ all of whose powers are rigid and $\sqrt{rD}$ is a prime ideal. Calling $D$ a semi packed domain (SPD) if every nonzero non unit of $D$ is a finite product of packed elements, we study SPDs and explore situations in which an SPD is a semirigid GCD domain.
