{"id":7898,"date":"2022-05-11T14:28:16","date_gmt":"2022-05-11T08:58:16","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7898"},"modified":"2025-06-02T20:46:19","modified_gmt":"2025-06-02T15:16:19","slug":"what-is-a-ring-in-discrete-mathematics","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-a-ring-in-discrete-mathematics\/","title":{"rendered":"What is a ring in discrete mathematics?"},"content":{"rendered":"\n

Definition<\/strong>(Ring)<\/h2>\n\n\n\n

A non-empty set R, equipped with two binary operations, called addition (+) and multiplication (.) is called a ring<\/a> if the following postulates are satisfied.<\/p>\n\n\n\n

(1) R is an additive Abelian group,<\/p>\n\n\n\n

(2) R is an multiplicative semigroup<\/p>\n\n\n\n

(3) The two distributive laws hold good, viz<\/p>\n\n\n\n

a. (b + c) =a.b + a.c. (left distributive law)<\/p>\n\n\n\n

(a + b). c = a.c + b.c, (right distributive law) for all a, b, c R<\/p>\n\n\n\n

Note: <\/strong><\/h2>\n\n\n\n

1. In order to be a ring<\/a>, there must be a non-empty set equipped with two binary operations, viz, + and . , satisfying the postulates mentioned above. Later on we will simply write \u2018\u2018Let R be a ring\u2019\u2019. It should be borne in mind that there are two binary operations, viz, + and , on R.<\/p>\n\n\n\n

2. It should also be borne in mind that the binary operations + and . may not be own usual addition and multiplication.<\/p>\n\n\n\n

3. Since R is an additive Abelian group, the additive identity, denoted by 0, belongs to R. It is called the zero element of R. Here 0 is a symbol.<\/p>\n\n\n\n

It should not be confused with the real number zero.<\/p>\n\n\n\n

4. If a , bR, then \u2013 bR ( R is an additive group). a+ (\u2013 b) is, generally, denoted by a \u2013 b.<\/p>\n\n\n\n

5. a.b is, generally, written as ab.<\/p>\n\n\n\n

Commutative Ring: <\/strong>R is commutative if ab = ba, for all a, bR.<\/p>\n\n\n\n

Ring with unity <\/strong>: R is a ring with unity (or identity) if there exists an element e R such that<\/p>\n\n\n\n

ae = a = ea, for all a R.<\/p>\n\n\n\n

Note<\/strong>: A ring may or may not have a unity.<\/p>\n\n\n\n

2. If R is a ring with unity, unity may be same as the zero element. For example, {0} is a commutative R with unity. Here 0 is the additive as well as the multiplicative identity. It is the zero ring.<\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n