Unique tensor factorization of loop-resistant algebras over a field of finite characteristic
By an algebra \(A\), the author means a finite-dimensional, associative, unitary, and split \(k\)-algebra. Split means that the residue algebra \(A/\text{rad }A\) is a sum of matrix algebras over the ground field \(k\). By \(A_1\oplus A_2\) is denoted the sum of algebras \(A_1\), \(A_2\) and by \(A_1\otimes A_2\) is denoted their tensor product. An algebra is said to be \(\otimes\)-indecomposable (\(\oplus\)-indecomposable) if it cannot be written as a proper product (sum). A finite-dimensional, \(\oplus\)-indecomposable algebra \(A\) is called loop-resistant if there is an idempotent \(e\in A\) with \(A/\langle 1-e\rangle=k^{n\times n}\) for some \(n\). A finite-dimensional algebra is said to be loop-resistant if each \(\oplus\)-indecomposable direct summand is loop-resistant. The main result established by the author is: Let \(k\) be a field. The set \(M\) of isomorphism classes of \(\oplus\)-indecomposable, loop-resistant, split \(k\)-algebras endowed with the operation induced by the tensor product is a free commutative monoid over the set \(\aleph\) of isomorphism classes of \(\{\oplus,\otimes\}\)-indecomposable, loop-resistant, split algebras. The semiring \(U\) of isomorphism classes of loop-resistant, split algebras is the positive cone \(\mathbb{N}[\aleph]\) of the polynomial ring \(\mathbb{Z}[\aleph]\).
scientific article; zbMATH DE number 1846012