COMMUTATIVE RING Meaning and
Definition
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A commutative ring is an algebraic structure that consists of a set equipped with two binary operations, addition and multiplication. The addition operation follows the rules of commutativity, meaning that the order in which elements are added together does not affect the result. In other words, for any two elements a and b in the set, a + b = b + a.
The multiplication operation also satisfies the commutative property, meaning that the order of multiplication does not alter the outcome. Specifically, for any two elements a and b in the set, a * b = b * a.
Additionally, a commutative ring must satisfy certain other properties. Firstly, it must contain an additive identity element, typically denoted as 0, such that for any element a, a + 0 = a. It must also possess additive inverses, which means that for every element a, there exists another element -a such that a + (-a) = 0.
Furthermore, the commutative ring must have a multiplicative identity element, often denoted as 1, which satisfies the property that for every element a, a * 1 = a. Lastly, the distributive property must hold, which states that for any three elements a, b, and c in the set, the product of a and the sum of b and c is equal to the sum of the products of a and b, and a and c. Symbolically, a * (b + c) = (a * b) + (a * c).
In summary, a commutative ring is a mathematical structure that exhibits commutativity in both addition and multiplication, while also fulfilling certain additional requirements.
Common Misspellings for COMMUTATIVE RING
- xommutative ring
- vommutative ring
- fommutative ring
- dommutative ring
- cimmutative ring
- ckmmutative ring
- clmmutative ring
- cpmmutative ring
- c0mmutative ring
- c9mmutative ring
- conmutative ring
- cokmutative ring
- cojmutative ring
- comnutative ring
- comkutative ring
- comjutative ring
- commytative ring
- commhtative ring
- commjtative ring
- commitative ring
Etymology of COMMUTATIVE RING
The term "commutative ring" is composed of two parts: "commutative" and "ring".
1. Commutative: The word "commutative" comes from the Latin word "commutare", which means "to change or exchange". The term "commutative" refers to the property of an operation that does not change regardless of the order of its operands. It was first used in mathematics by the French mathematician Augustin-Louis Cauchy in the early 19th century.
2. Ring: The word "ring" originated from the Old English word "hring", which means "a circular object". It can be traced back to the Proto-Germanic word "*hringaz". In mathematics, the concept of a ring refers to a set equipped with two binary operations, addition and multiplication, satisfying certain properties.
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