Let R be a ring with identity and let S be a subring of R containing the identity. Prove that if m is a unit in S then u is a unit in R. Show by example that the converse is false.

Question

06-12-2019
Mathematics

contestada

Let R be a ring with identity and let S be a subring of R containing the identity. Prove that if m is a unit in S then u is a unit in R. Show by example that the converse is false.

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EnriquePires EnriquePires · Answer 1 · 2019-12-11T00:04:49+01:00

Explanation:

Since m is a unit in S, then, there exists b ∈ S such that m*b = 1, where 1 is the identity. Since S is a subring of R we have that m ∈ R, and therefore b is also the multiplicative inverse of m in R. The converse isnt true.

The set of real numbers is a Ring with the standard sum and multiplication. Every real number different from 0 has a multiplicative inverse. For example, the inverse of 2 is 1/2. However, 2 is not a unit on the subring of Integers Z.