On ideal and subalgebra coefficients in a class k-algebras
Abstract
Let k be a commutative field with prime field 
and A a k- algebra. Moreover, assume that there is a k-vector space basis 
of A that satisfies the following condition: for all 
,the product 
is contained in the -vector space spanned by . 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 
with U-as one-sided ideal resp. as k-algebra—being generated by elements in the 
-vector space spanned by .
and A a k- algebra. Moreover, assume that there is a k-vector space basis
of A that satisfies the following condition: for all ,the product is contained in the -vector space spanned by . 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 with U-as one-sided ideal resp. as k-algebra—being generated by elements in the -vector space spanned by .DOI Code:
10.1285/i15900932v27n1p77
Keywords:
Field of definition; Non-associative k-algebra; One-sided ideal; k-subalgebra
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