Principal Ideals

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Abstract Algebra › Principal Ideals

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Which of the following is an ideal of a ring?

Maximal Ideal

CORRECT

Communicative Ideal

Minimal Ideal

Associative Ideal

None are ideals

Explanation

When dealing with rings there are three main ideals

Proper Ideal: When is a commutative ring, and is a non empty subset of then, is said to have a proper ideal if both the following are true.

$a\pm b \ \epsilon\ A\ \forall\ a,b\ \epsilon\ I$ and $ca\ \epsilon\ A,\ a\ \epsilon\ A, c\ \epsilon\ R$

Prime Ideal: When is a commutative ring, is a prime ideal if

$\forall\ a,b\ \epsilon\ R$ is true and $ab\ \epsilon\ A \rightarrow a\ \epsilon\ A \vee\ b\ \epsilon\ A$

Maximal Ideal: When is a commutative ring, and is a non empty subset of then, has a maximal ideal if all ideal are

$\A\subseteq B\subseteq R, \ B=A \vee\ B=R$

Looking at the possible answer selections, Maximal Ideal is the correct answer choice.

Which of the following is an ideal of a ring?

Prime Ideal

CORRECT

Minimum Ideal

Associative Ideal

Multiplicative Ideal

All are ideals of rings.

Explanation

When dealing with rings there are three main ideals

Proper Ideal: When is a commutative ring, and is a non empty subset of then, is said to have a proper ideal if both the following are true.

$a\pm b \ \epsilon\ A\ \forall\ a,b\ \epsilon\ I$ and $ca\ \epsilon\ A,\ a\ \epsilon\ A, c\ \epsilon\ R$

Prime Ideal: When is a commutative ring, is a prime ideal if

$\forall\ a,b\ \epsilon\ R$ is true and $ab\ \epsilon\ A \rightarrow a\ \epsilon\ A \vee\ b\ \epsilon\ A$

Maximal Ideal: When is a commutative ring, and is a non empty subset of then, has a maximal ideal if all ideal are

$\A\subseteq B\subseteq R, \ B=A \vee\ B=R$

Looking at the possible answer selections, Prime Ideal is the correct answer choice.

Which of the following is an ideal of a ring?

Proper Ideal

CORRECT

Minimal Ideal

Associative Ideal

Communicative Ideal

All are ideals

Explanation

When dealing with rings there are three main ideals

Proper Ideal: When is a commutative ring, and is a non empty subset of then, is said to have a proper ideal if both the following are true.

$a\pm b \ \epsilon\ A\ \forall\ a,b\ \epsilon\ I$ and $ca\ \epsilon\ A,\ a\ \epsilon\ A, c\ \epsilon\ R$

Prime Ideal: When is a commutative ring, is a prime ideal if

$\forall\ a,b\ \epsilon\ R$ is true and $ab\ \epsilon\ A \rightarrow a\ \epsilon\ A \vee\ b\ \epsilon\ A$

Maximal Ideal: When is a commutative ring, and is a non empty subset of then, has a maximal ideal if all ideal are

$\A\subseteq B\subseteq R, \ B=A \vee\ B=R$

Looking at the possible answer selections, Prime Ideal is the correct answer choice.

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