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Ring with unity

 Def. Ring with unity : 

A ring R ¹ {0} is said to be a ring with unity if it contains multiplicative identity i.e., there exists an element denoted by 1Î R such that a. 1= a ==  1.a  for all a  ÎR . Def. Division ring or Skew-field : A ring R is said to be a division ring or skew-field if

(i)                  R is with unity i.e. 1Î R .

(ii)                Every non-zero element of R has a multiplicative inverse i.e., for every non-zero element a  ÎR there exists an element b  Î Rsuch that a. b= b. a  =1. We usually denote multiplicative inverse of a by a-1


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