On zero product determined algebras
On zero product determined algebras
复制标题
DOI：
10.1080/03081087.2013.866668
复制
发表时间：
2015-02
影响因子：
1.1
通讯作者：
Daniel Brice;Huajun Huang
中科院分区：
文献类型：
--
作者：
Daniel Brice;Huajun Huang
设一个有恒等的交换环。我们说-代数是零积，它决定了如果对于每一个-双线性，具有这样的性质，无论何时，都存在一个-线性，使得对所有。给出了代数为零积确定的一个充分必要条件，并利用该条件导出了几个新的结果。其中，我们证明了代数的直接和是零积确定的当且仅当各分量代数是零积确定时；我们证明了零积确定的代数的张量积是零积确定的如果是场或者代数乘法是满射；给出了零积确定代数的同态象为零积确定的条件；最后，我们引入了一类零积确定矩阵代数，它推广了块上三角矩阵，并推广了Brešar， Grašič， and Ortega（2009）的结果。
Let be a commutative ring with identity. A -algebra is said to be zero product determined if for every -bilinear having the property that whenever , there is a -linear such that for all . We provide a necessary and sufficient condition for an algebra to be zero product determined and use the condition to derive several new results. Among these, we show that the direct sum of algebras is zero product determined if and only if each component algebra is zero product determined; we show that the tensor product of zero product determined algebras is zero product determined in case is a field or in case the algebra multiplications are surjective; we produce conditions under which the homomorphic images of a zero product determined algebra are zero product determined; finally, we introduce a class of zero product determined matrix algebras that generalizes block upper triangular matrices and extends a result of Brešar, Grašič, and Ortega in 2009.