Definition:Inverse (Abstract Algebra)/Inverse

Definition

Let $\left({S, \circ}\right)$ be a monoid whose identity is $e_S$.

An element $y \in S$ such that $y \circ x = e_S = x \circ y$, that is, $y$ is both a left inverse and a right inverse of $x$, then $y$ is a two-sided inverse (or simply inverse) of $x$.

The notation used to represent an inverse of an element depends on the set and binary operation under consideration.

Various symbols are seen for a general inverse, for example $\hat x$ and $x^*$.

In multiplicative notation:

If $s \in S$ has an inverse, it is denoted $s^{-1}$.

If the operation concerned is commutative, then additive notation is often used:

If $s \in S$ has an inverse, it is denoted $-s$.

Also see

• Left Inverse
• Right Inverse

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.4$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 4$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967)... (previous)... (next): $\text{II}$: The Group Property
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970)... (previous)... (next): $\S 1.1$: The definition of a ring: Definitions $1.1 \ \text{(b)}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 27$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 31$