Structure Induced by Ring with Unity Operations is Ring with Unity

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Theorem

Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.

Let $S$ be a set.

Let $\struct {R^S, +', \circ'}$ be the structure on $R^S$ induced by $+'$ and $\circ'$.

Then $\struct {R^S, +', \circ'}$ is a ring with unity $f_{1_R}:S \to R$ defined by:

$\forall s \in S: f_{1_R} \paren{s} = 1_R$

Proof

By Structure Induced by Ring Operations is Ring then $\struct {R^S, +', \circ'}$ is a ring.

We have from Induced Structure Identity that the constant mapping $f_{1_R}: S \to R$ defined as:

$\forall x \in S: f_{1_R} \paren {x} = 1_R$

is the identity for $\struct {R^S, \circ'}$.

The result follows by definition of ring with unity.

$\blacksquare$


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