Test for Ideal

Theorem

Let $J$ be a subset of a ring $\struct {R, +, \circ}$.

Then $J$ is an ideal of $\struct {R, +, \circ}$ if and only if these all hold:

$(1): \quad J \ne \O$

$(2): \quad \forall x, y \in J: x + \paren {-y} \in J$

$(3): \quad \forall j \in J, r \in R: r \circ j \in J, j \circ r \in J$

Let $J$ be an ideal of $\struct {R, +, \circ}$.

Then conditions $(1)$ to $(3)$ hold by virtue of the ring axioms and $J$ being an ideal.

$\Box$

Suppose conditions $(1)$ to $(3)$ hold.

As $r \in J \implies r \in R$, if $(3)$ holds for $J$, then $J$ is closed under $\circ$ and condition $(3)$ of Subring Test holds.

Thus, $J$ is a subring of $R$.

As $(3)$ defines the condition for $J$, being a subring, to be an ideal, the result holds.

So $J$ is an ideal of $\struct {R, +, \circ}$.

$\blacksquare$

1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals: Theorem $34$
1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms: Definition $2.5$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 58.4$ Ideals