Rng

Remarks

No multiplicative identity is required; a ring is a rng that has one. Other names are non-unital ring and pseudo-ring. Example: the even integers $2\Z$ with the usual operations form a rng; the clauses hold because they hold in $\Z$ and $2\Z$ is closed under both operations. It has no identity, since $e \cdot 2 = 2$ forces $e = 1 \notin 2\Z$. Non-example: $\R^3$ with the cross product as multiplication fails clause 2. With the standard basis $i, j, k$, one has $(i \times i) \times j = 0$ but $i \times (i \times j) = i \times k = -j$. Every rng $R$ embeds as an ideal in the ring $\Z \times R$ with the product $(m, a)(n, b) = (mn, mb + na + ab)$, whose identity is $(1, 0)$.