# Homotopy Type Theory commutative discrete reciprocal ring > history (changes)

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## Definition

A

discrete reciprocal ring is a ring $(A, +, -, 0, \cdot, 1)$ with a commutative identity for $\cdot$:

$m_\kappa:\prod_{(a:A)} \prod_{(b:A)} a\cdot b = b\cdot a$

## Properties

Every commutative discrete reciprocal ring is a commutative discrete division ring.