I know that a subring $R'$ of a ring $R$ is generated by $S\subseteq R$ if $R'$ is equal with the intersection of all subrings of $R$ those contains $S$. I want to know there is and easy way to write the elements of $R'$ in terms of the elements of $S$? Like what occurs for generated ideals in a ring.
Ring generated by $S$
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abstract-algebra
ring-theory
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1This is simpler/easier to do if $R$ is a commutative ring. Please clarify whether $R$ is commutative and whether is has unity (multiplicative identity). – 2017-02-10
1 Answers
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Let $A$ be the set of all finite sums and differences of finite products of elements of $S$. Then $A$ is clearly a ring and contains $S$ and is contained in any ring that contains $S$. We conclude that $A=R'$.