We have defined a new finite-dimensional commutative algebra, which is proved semisimple. Let us call it t-algebra. Due to its semisimplicity and commutativity, the t-algebra is a direct product of a finite number of algebraic constituents. These algebraic constituents are all subsets of the t-algebra and are themselves algebras. However, the multiplicative identity of each constituent is not the same as the identity of the t-algebra. More precisely, the identity of the t-algebra does not belong to any of those mentioned above "sub-algebra". Each of the "subalgebras" has its unique multiplicative identity.
By definition, albeit an algebra itself, each constituent mentioned above is NOT a subalgebra of t-algebra. Then comes my question. What is the name of a “sub-algebra” without the identity of its super-algebra? Do they have a formal name? Can I coin a name such as quasi-sub-algebra for them?
They’re all ideals and also all quotients. In this particular case I would just call them the “factors.” If you really want to call them subalgebras then maybe “non-unital subalgebras” but I would avoid this personally.