Test for Ideal
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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:
- $(1): \quad J \ne \O$
- $(2): \quad \forall x, y \in J: x + \paren {-y} \in J$
- $(3): \quad \forall j \in J, r \in R: r \circ j \in J, j \circ r \in J$
Proof
Necessary Condition
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.
$\Box$
Sufficient Condition
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 $\struct {R, +, \circ}$.
$\blacksquare$
Sources
- 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