Abstract
AbstractA basis ${\mathcal{B}}=\{u_{i}\}_{i\in I}$ of a commutative or anticommutative algebra $\mathfrak{C},$ over an arbitrary base field $\mathbb{F}$, is called multiplicative if for any $i,j\in I$ we have that $u_{i}u_{j}\in \mathbb{F}u_{k}$ for some $k\in I$. We show that if a commutative or anticommutative algebra $\mathfrak{C}$ admits a multiplicative basis then it decomposes as the direct sum $\mathfrak{C}=\bigoplus _{j}\mathfrak{i}_{j}$ of well-described ideals each one of which admits a multiplicative basis. Also the minimality of $\mathfrak{C}$ is characterised in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is indexed by the family of its minimal ideals admitting a multiplicative basis.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.