\section{Negative of Subring is Negative of Ring}
Tags: Ring Theory

\begin{theorem}
Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
For each $x \in R$ let $-x$ denote the [[Definition:Ring Negative|ring negative]] of $x$ in $R$.
Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a [[Definition:Subring|subring]] of $R$.
For each $x \in S$ let $\mathbin \sim x$ denote the [[Definition:Ring Negative|ring negative]] of $x$ in $S$.
Then:
:$\forall x \in S: \mathbin \sim x = -x$
\end{theorem}

\begin{proof}
Let $i_S: S \to R$ be the [[Definition:Inclusion Mapping|inclusion mapping]] from $S$ to $R$.
By [[Inclusion Mapping on Subring is Homomorphism]], $i_S$ is a [[Definition:Ring Homomorphism|ring homomorphism]].
By [[Ring Homomorphism of Addition is Group Homomorphism]], $i_S$ is a [[Definition:Group Homomorphism|group homomorphism]] on [[Definition:Ring Addition|ring addition]] $+$.
Let $x \in S$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbin \sim x
      | r = \map {i_S} {\mathbin \sim x}
      | c = as $\mathbin \sim x \in S$
}}
{{eqn | r = -x
      | c = [[Group Homomorphism Preserves Inverses]]
}}
{{end-eqn}}
{{qed}}
[[Category:Ring Theory]]
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\end{proof}


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