An associative algebra refers to a mathematical structure consisting of a set of elements along with operations of addition and multiplication that satisfy certain criteria. There are various synonyms for the term 'associative algebra', which includes ring, algebraic system, algebras, algebraic structure, and algebraic object. A ring is a general term referring to a mathematical object that possesses two binary operations of addition and multiplication with some specific properties, while an algebraic system can broadly be defined as any set of mathematical objects with associated operations and properties. Algebraic structure denotes a set along with one or more operations and can be either associative or non-associative, whereas the term algebraic object typically refers to a class of algebras satisfying some specific axioms.