Null Ring is Subring of Ring

Theorem

Let $R$ be a ring.

Then the null ring is a subring of $R$.

Proof

From Null Ring is Ring, the null ring $\struct {\set {0_R}, +, \circ}$ is a ring.

We also have that $\set {0_R}$ is a subset of $R$

Hence the result by definition of subring.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 19$. Subrings: Example $30$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 56.1$ Subrings and subfields