6.1. A RECOLLECTION OF RINGS 267 We gave a number of examples of subrings in Example 1.11.5 . You are asked to verify these examples, and others, in the Exercises. For any ring R and any subset S ± R there is a smallest subring of R that contains S , which is called the subring generated by S . We say that R is generated by S as a ring if no proper subring of R contains S . A “constructive” view of the subring generated by S is that it consists of all possible ﬁnite sums of ﬁnite products ˙ T 1 T 2 ²²² T n , where T i 2 S . In particular, the subring generated by a single element T 2 R is the set of all sums P n i D 1 n i T i . (Note there is no term for i D0 .) The subring generated by T and the multiplicative identity 1 (assuming that R has a multiplicative identity) is the set of all sums n0 1 C P n i D 1 n i T i D P n i D0 n i T i , where we use the convention T0 D 1 . The subring generated by S is equal to the intersection of the family of all subrings of R that contain S ; this family is nonempty since
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