Modulo Multiplication has Identity

Theorem

Multiplication modulo $m$ has an identity:

$\forall \eqclass x m \in \Z_m: \eqclass x m \times_m \eqclass 1 m = \eqclass x m = \eqclass 1 m \times_m \eqclass x m$

Follows directly from the definition of multiplication modulo $m$:

$\ds \eqclass x m \times_m \eqclass 1 m$

$=$

$\ds \eqclass {x \times 1} m$

$\ds $

$=$

$\ds \eqclass x m$

$\ds $

$=$

$\ds \eqclass {1 \times x} m$

$\ds $

$=$

$\ds \eqclass 1 m \times_m \eqclass x m$

Thus $\eqclass 1 m$ is the identity for multiplication modulo $m$.

$\blacksquare$

1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.11$
1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Theorem $5$
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Examples of Group Structure: $\S 34$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 19.1$: Properties of $\Z_m$ as an algebraic system