# Definition:Product Inverse

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## Definition

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

Let $U_R$ denotes the group of units of $R$.

The inverse of $x \in U_R$ by $\circ$ is called the **(ring) product inverse of $x$**.

The usual means of denoting the product inverse of an element $x$ is by $x^{-1}$.

Thus it is distinguished from the additive inverse of $x$, that is, the (ring) negative of $x$, which is usually denoted $-x$.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $4$: Fields: $\S 14$. Definition of a Field