在环论中,幂等元是很重要的一类元素,幂等元是指满足a2=a的元素a。任何含单位元的环通常都有两个幂等元,即0和1,这两个特殊的幂等元通常被称为平凡幂等元。然而,在环ℤn和ℤn[ x ]中,可能存在非平凡幂等元。本文将研究多项式环ℤp2q2[ x ]...在环论中,幂等元是很重要的一类元素,幂等元是指满足a2=a的元素a。任何含单位元的环通常都有两个幂等元,即0和1,这两个特殊的幂等元通常被称为平凡幂等元。然而,在环ℤn和ℤn[ x ]中,可能存在非平凡幂等元。本文将研究多项式环ℤp2q2[ x ]中的幂等元,并进一步探究多项式环ℤp2q2[ x ]上的2阶矩阵环M2(ℤp2q2[ x ])中非平凡幂等元的形式与性质,其中p、q为不同素数。研究结果表明,ℤp2q2[ x ]中有4个幂等元,M2(ℤp2q2[ x ])中有7个非平凡幂等矩阵。记环R的幂等元集合为Id(R)。展开更多
In this paper,potent index of an element and pseudo clean rings are considered.Some properties and examples of pseudo clean rings are given.We also show that Zm is pseudo clean for every 2≤m∈Z and pseudo clean rings...In this paper,potent index of an element and pseudo clean rings are considered.Some properties and examples of pseudo clean rings are given.We also show that Zm is pseudo clean for every 2≤m∈Z and pseudo clean rings are clean.Furthermore,we prove pseudo clean rings are directly finite and have stable range one.展开更多
文摘在环论中,幂等元是很重要的一类元素,幂等元是指满足a2=a的元素a。任何含单位元的环通常都有两个幂等元,即0和1,这两个特殊的幂等元通常被称为平凡幂等元。然而,在环ℤn和ℤn[ x ]中,可能存在非平凡幂等元。本文将研究多项式环ℤp2q2[ x ]中的幂等元,并进一步探究多项式环ℤp2q2[ x ]上的2阶矩阵环M2(ℤp2q2[ x ])中非平凡幂等元的形式与性质,其中p、q为不同素数。研究结果表明,ℤp2q2[ x ]中有4个幂等元,M2(ℤp2q2[ x ])中有7个非平凡幂等矩阵。记环R的幂等元集合为Id(R)。
基金Supported by National Natural Science Foundation of China(12301041)。
文摘In this paper,potent index of an element and pseudo clean rings are considered.Some properties and examples of pseudo clean rings are given.We also show that Zm is pseudo clean for every 2≤m∈Z and pseudo clean rings are clean.Furthermore,we prove pseudo clean rings are directly finite and have stable range one.
