On the Inner Automorphisms of Tensor Spaces and Algebraic Extensions

We study algebraic structures on tensor spaces $T = F^{m_1 \times \cdots \times m_n}$ over fields or rings $F$, where operations are defined component-wise on tensor entries. Our central result establishes that when scalar operations on $F$ satisfy the axioms of an algebraic structure (group, ring, field, module, or vector space), the corresponding entrywise operations endow $T$ with the same structure. This preservation is complete: $(T,+)$ forms an abelian group when $(F,+)$ does, $(T,+,\star)$ is a ring when $(F,+,\star)$ is a ring, and $T$ becomes an $R$-module when $F$ is an $R$-module under entrywise scalar action. Index permutations $\pi \in \Sym(I)$ induce automorphisms $P_\pi$ of these tensor algebras, formalizing symmetries without altering tensor dimensions. We characterize when $T$ can be a field (only for single-coordinate tensors) and describe the unit group structure. The study applies naturally to many fields, however we have investigate its application in neural network weight tensors.