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Abstract Algebra › Abstract Algebra

Questions 1 - 10

Which of the following is an ideal of a ring?

Maximal Ideal

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?

Maximal Ideal

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.

Identify the following definition.

Given that lives in the Euclidean plane $\mathbb{R}^2$ . Elements , , and in the subfield that form a straight line who's equation form is , is known as a .

Line in

Circle in

Plane

Angle

Subfield

Explanation

By definition, given that lives in the Euclidean plane $\mathbb{R}^2$ . When elements , , and in the subfield , form a straight line who's equation form is , is known as a line in .

Which of the following is an identity element of the binary operation ?

Explanation

Defining the binary operation will help in understanding the identity element. Say is a set and the binary operator is defined as $A\times A\rightarrow A$ for all given pairs in .

Then there exists an identity element in such that given,

$\a,b,c \ \epsilon\ A$

$\a*e=a, e*a=a \b*e=b, e*b=b \c*e=c,e*c=c$

Therefore, looking at the possible answer selections the correct answer is,

Which of the following is an identity element of the binary operation ?

Explanation

Defining the binary operation will help in understanding the identity element. Say is a set and the binary operator is defined as $A\times A\rightarrow A$ for all given pairs in .

Then there exists an identity element in such that given,

$\a,b,c \ \epsilon\ A$

$\a*e=a, e*a=a \b*e=b, e*b=b \c*e=c,e*c=c$

Therefore, looking at the possible answer selections the correct answer is,

Which of the following is an identity element of the binary operation ?

Explanation

Defining the binary operation will help in understanding the identity element. Say is a set and the binary operator is defined as $A\times A\rightarrow A$ for all given pairs in .

Then there exists an identity element in such that given,

$\a,b,c \ \epsilon\ A$

$\a*e=a, e*a=a \b*e=b, e*b=b \c*e=c,e*c=c$

Therefore, looking at the possible answer selections the correct answer is,

identify the following definition.

Given is a normal subgroup of , it is denoted that when the group of left cosets of in is called .

Factor Group

Simple Group

Subgroup

Normal Group

Cosets

Explanation

By definition of a factor group it is stated,

Given is a normal subgroup of , it is denoted that when the group of left cosets of in is called the factor group of which is determined by .

Which of the following is an ideal of a ring?

Maximal Ideal

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.

Identify the following definition.

Given that lives in the Euclidean plane $\mathbb{R}^2$ . Elements , , and in the subfield that form a straight line who's equation form is , is known as a .

Line in

Circle in

Plane

Angle

Subfield

Explanation

By definition, given that lives in the Euclidean plane $\mathbb{R}^2$ . When elements , , and in the subfield , form a straight line who's equation form is , is known as a line in .

identify the following definition.

Given is a normal subgroup of , it is denoted that when the group of left cosets of in is called .

Factor Group

Simple Group

Subgroup

Normal Group

Cosets