Definition:Maximal Ideal of Ring
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Definition
Let $R$ be a ring.
An ideal $J$ of $R$ is maximal if and only if:
- $(1): \quad J \subsetneq R$
- $(2): \quad$ There is no ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.
That is, if and only if $J$ is a maximal element of the set of all proper ideals of $R$ ordered by the subset relation.
Maximal Left Ideal
A left ideal $J$ of $R$ is a maximal left ideal if and only if:
- $(1): \quad J \subsetneq R$
- $(2): \quad$ There is no left ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.
Maximal Right Ideal
A right ideal $J$ of $R$ is a maximal right ideal if and only if:
- $(1): \quad J \subsetneq R$
- $(2): \quad$ There is no right ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.
It follows that in a commutative ring, a maximal left ideal, a maximal right ideal and a maximal ideal are the same thing.
Also defined as
Some sources insist that $R$ be a commutative ring with unity for this definition to hold.
Also see
- Results about maximal ideals of rings can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 23$. Maximal Ideals
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2$: Exercise $12$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 63$. Construction of fields as factor rings
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.8$ Algebraic properties of $p$-adic integers