An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra.
A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to associative algebras or to Lie algebras. Only for unital algebras is there a stronger notion, of unital subalgebra, for which it is also required that the unit of the subalgebra be the unit of the bigger algebra.
The 2×2-matrices over the reals form a unital algebra in the obvious way. The 2×2-matrices for which all entries are zero, except for the first one on the diagonal, form a subalgebra. It is also unital, but it is not a unital subalgebra.
I think, the question consists not in exemplifying such algebras but in precise describing the class of such algebras. But before we will try to give answer we have to make more precise the question. Dear @Lerato_Mashego, could you tell us what you are asking about?
- Badges
- Science topic
- Similar topics
- Mathematics
- Algebra
- Matrix