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In Dummit and Foote Abstract ALgebra, they define

A subring of the ring $R$ is a subgroup of $R$ that is closed under multiplication.

I know that since we check for closure and multiplication and subtraction, it is actually closed under addition, additive inverse and multiplication.

I don't understand how being closed under addition and additive inverses is in the defintion, does subgroup imply a subgroup under addition, even though they didn't specify the group law for the subgroup? Are subgroups implicitly groups under addition when talking about rings?

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Yes, a subgroup of $R$ is indeed a subgroup of $(R,+)$, you are right. The reason is, that $(R^{\times},\cdot)$ is not a group (except for fields). Take $R=\Bbb Z$ the integers. It doesn't make sense to take about the group $(\Bbb Z^{\times},\cdot)$ under multiplication.

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