On ideal and subalgebra coefficients in a class k-algebras


Let k be a commutative field with prime field k<sub>0</sub> and A a k- algebra. Moreover, assume that there is a k-vector space basis ω of A that satisfies the following condition: for all ω<sub>1</sub>, ω<sub>2</sub> ∈ ω ,the product ω<sub>1</sub>ω<sub>2</sub> is contained in the k<sub>0</sub>-vector space spanned by ω. It is proven that the concept of minimal field of definition from polynomial rings and semigroup algebras can be generalized to the above class of (not necessarily associative) k-algebras. Namely, let U be a one-sided ideal in A or a k-subalgebra of A. Then there exists a smallest k' ≤ k with U-as one-sided ideal resp. as k-algebra—being generated by elements in the k'-vector space spanned by ω.

DOI Code: 10.1285/i15900932v27n1p77

Keywords: Field of definition; Non-associative k-algebra; One-sided ideal; k-subalgebra

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