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Abstract Algebra : Rings

Study concepts, example questions & explanations for Abstract Algebra

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Example Questions

Example Question #1 : Principal Ideals

Which of the following is an ideal of a ring?

Possible Answers:

Multiplicative Ideal

All are ideals of rings.

Associative Ideal

Prime Ideal

Minimum Ideal

Correct answer:

Prime Ideal

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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Example Question #2 : Principal Ideals

Which of the following is an ideal of a ring?

Possible Answers:

Associative Ideal

Communicative Ideal

None are ideals

Minimal Ideal

Maximal Ideal

Correct answer:

Maximal Ideal

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.

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Example Question #3 : Principal Ideals

Which of the following is an ideal of a ring?

Possible Answers:

Associative Ideal

All are ideals

Minimal Ideal

Proper Ideal

Communicative Ideal

Correct answer:

Proper Ideal

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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Abstract Algebra : Rings

Study concepts, example questions & explanations for Abstract Algebra

All Abstract Algebra Resources

Example Questions

Example Question #1 : Principal Ideals

Example Question #2 : Principal Ideals

Example Question #3 : Principal Ideals

All Abstract Algebra Resources

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