On the Conditions Under Which an Algebra over a Field Has a Multiplicative Identity Element

The set of all annihilators in a ring forms a lattice whose structure reflects the internal properties of the ring. In contrast to the lattices of one-sided ideals, lattices of one-sided annihilators possess a particularly desirable property: symmetry. More precisely, the lattice of left annihilators in a given ring is anti-isomorphic to the lattice of right annihilators. In the literature, attention is primarily focused on rings with identity. In this work, we study lattices of one-sided annihilators in rings without a multiplicative identity element. The properties of rings with an identity element usually differ significantly from those of rings without an identity. We restrict our consideration to rings that are algebras over fields. We indicate certain connections between algebras A without identity and their extensions to algebras A1 with identity (we consider the Dorroh extension). We highlight similarities and differences in the structure of the lattices of one-sided annihilators in A and those in A1. We also identify conditions on annihilators in semiprimary algebras that ensure the existence of an identity element. Furthermore, we show that the existence of an identity in an algebra A is not guaranteed even if A satisfies the condition A2=A and possesses a one-sided identity.