Field has 2 Ideals

Theorem

Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Then the only ideals of $\struct {F, +, \circ}$ are $\struct {F, +}$ and $\set {0_F}$.


That is, $\struct {F, +, \circ}$ has no non-null proper ideals.


Proof

By definition, a field is a division ring.

From Null Ring is Ideal and Ring is Ideal of Itself, it is always the case that $\set {0_F}$ and $\struct {F, +}$ are ideals of $\struct {F, +, \circ}$.

From Ideals of Division Ring, it follows that the only ideals of $\struct {F, +, \circ}$ are $\struct {F, +}$ and $\set {0_F}$.

$\blacksquare$


Sources

  • 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals: Theorem $36$
  • 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 58.3$ Ideals
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