A Study on Structures and Invariants of Rings: Exploring the properties and variations of rings in algebraic structures

Abstract

In mathematics, aring is an algebraic structure consisting of a set together with two binaryoperations usually called addition and multiplication, where the set is anabelian group under addition (called the additive group of the ring) and amonoid under multiplication such that multiplication distributes over addition.  In other words the ring axioms require thataddition is commutative, addition and multiplication are associative,multiplication distributes over addition, each element in the set has anadditive inverse, and there exists an additive identity. One of the most commonexamples of a ring is the set of integers endowed with its natural operationsof addition and multiplication. Certain variations of the definition of a ringare sometimes employed, and these are outlined later in the article.