Ideal Theory in Ring Theory

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Questions and Answers

If I and J are two ideals of a ring R, what can be said about their product IJ?

It is equal to I + J.

It is guaranteed to be an ideal of R. (correct)

It is always a subring of R.

It may not be an ideal of R.

What is true about the intersection of two subrings of a ring R?

It may not be a subring of R.

It is always a subring of R. (correct)

It is always an ideal of R.

It is never a subring of R.

Which statement is incorrect regarding ideals and subrings?

Every ideal of a ring is a subring.

The sum of two ideals is an ideal.

Union of two ideals need not be an ideal.

A subring must have the same unity as the ring. (correct)

What type of ideal is established when an ideal is maximal?

It is a prime ideal. Signup and view all the answers

In a ring with unity, what happens if an ideal I contains the unity element 1?

I is equal to R. Signup and view all the answers

Which property holds true for subrings of an integral domain?

They are guaranteed to be rings without zero divisors. Signup and view all the answers

Which of the following statements about characteristics of rings is true?

If S and R have the same unity, then char S = char R. Signup and view all the answers

What does a nilpotent element in a commutative ring indicate?

It does not contribute non-zero elements to the ring. Signup and view all the answers

Which of the following defines the structure of ℚ(√d)?

It includes elements of rational numbers and square roots of rational numbers. Signup and view all the answers

What can be concluded about the union of two ideals of a ring R?

It may or may not be an ideal. Signup and view all the answers

How is a ring homomorphism characterized based on the function 𝜙(𝑎) = 𝑎²?

It preserves the addition operation within the ring. Signup and view all the answers

Which statement accurately describes the characteristics of ideals in a commutative ring?

Every left ideal and right ideal must be an ideal. Signup and view all the answers

Which condition must be satisfied for an ideal P to be considered a prime ideal in a ring?

If $ab ext{ in } P$, then either $a ext{ in } P$ or $b ext{ in } P$. Signup and view all the answers

What does it mean for a rational number d to be square-free?

It is not divisible by the square of any prime number. Signup and view all the answers

What structure does the set of nilpotent elements form in a ring?

An ideal. Signup and view all the answers

Which theorem relates to the condition of ideals within a ring based on their intersections?

The Chinese Remainder Theorem. Signup and view all the answers

What is a characteristic of a ring where the operation + forms an abelian group?

It has an associate relation that partitions the ring. Signup and view all the answers

Which statement correctly defines a zero divisor in a ring?

It is a non-zero element such that the product with another non-zero element yields zero. Signup and view all the answers

What distinguishes an integral domain from a field?

Every non-zero element in a field has a multiplicative inverse. Signup and view all the answers

Which of the following correctly describes a trivial ring?

It consists only of the element zero. Signup and view all the answers

How is an idempotent element defined in a ring?

An element that equals itself when squared. Signup and view all the answers

In the context of rings, which statement about nilpotent elements is correct?

Each nilpotent element must satisfy $a^n = 0$ for some $n$ in natural numbers. Signup and view all the answers

What is the definition of a zero ring?

A ring where multiplication of any two elements yields zero. Signup and view all the answers

Which of the following statements is true regarding a semi-group?

It must satisfy an associative property under its defined operation. Signup and view all the answers

In a ring R, when is an element b classified as not a unit?

If $N(a) < N(ab)$ Signup and view all the answers

What concept is implied by the uniqueness of the decomposition of a non-zero, non-unit element a in R?

There is a one-to-one correspondence between the prime factors of elements. Signup and view all the answers

Which of the following statements is true regarding normal forms in the context of ED and UFD?

All elements in a UFD can be factored uniquely into primes. Signup and view all the answers

What is the primary characteristic of a field extension K/F?

K must have a dimension over F. Signup and view all the answers

Which of the following is NOT a property of the ring ℤ[√-n] for n > 3?

It is recognized as a UFD. Signup and view all the answers

What conclusion can be drawn from the relation ED ⇒ P.I.D. ⇒ U.F.D.?

All ED rings are also UFDs. Signup and view all the answers

Under what conditions is the quadratic ring ℤ[√D] isomorphic to a subfield of K?

If D is less than 0. Signup and view all the answers

What does the notation [K : F] represent in the context of field extensions?

The dimension of K over F. Signup and view all the answers

What is the smallest positive integer n called if it satisfies n a = 0 for all a in a ring R?

Characteristic Signup and view all the answers

In a commutative ring with unity (CRU), how is an element defined as irreducible?

If it cannot be expressed as a product of two non-unit elements. Signup and view all the answers

Which of the following represents the identity element in a ring R with respect to multiplication?

1 Signup and view all the answers

What is a factor of an element a in a commutative ring R?

An element b such that there exists c in R with b = ac. Signup and view all the answers

Which of the following statements describes a unit element in a ring with unity?

It is an element that has a multiplicative inverse in the ring. Signup and view all the answers

What defines a commutative ring (CR)?

The operation '∙' is commutative. Signup and view all the answers

If no positive integer n exists such that n a = 0 for all a in R, how is the characteristic of the ring R denoted?

Zero Signup and view all the answers

In a ring with unity, what is the major distinction between an improper factor and a proper factor?

Improper factors are units; proper factors are non-units. Signup and view all the answers

Study Notes

Ideals and Homomorphisms

Ideals (I) and (J) in a ring (R) can be represented with the mapping (\phi: R \to R) defined by (\phi(a) = a^2), which is a ring homomorphism.
If (P) is an ideal of ring (R), (R/P) is an integral domain if and only if (P) is a prime ideal.
Ideals can be expressed in the form (I + J) and (I \cap J), confirming their structural relationships in (R).

Square-Free and Quadratic Fields

A number (d) is termed square-free if not divisible by the square of any prime number.
For a square-free number (d), the quadratic field (\mathbb{Q}(\sqrt{d})) is defined as ({a + b\sqrt{d} : a, b \in \mathbb{Q}}), which forms a subfield of (\mathbb{C}).

Chinese Remainder Theorem

The theorem involves ideals (A_1, A_2, \ldots, A_k) in a ring (R), providing insights on factorization and congruences.

Properties of Rings and Ideals

Every left or right ideal in a commutative ring is an ideal, and subrings of commutative rings are themselves commutative.
The arbitrary intersection of ideals is also an ideal.
The sum of two ideals (I) and (J) is defined as (I + J = {a + b : a \in I, b \in J}) and is an ideal itself.
Zero divisors exist in rings, where non-zero elements (a) and (b) exist such that (ab = 0).

Definitions and Concepts

A ring is trivial if it contains only the zero element.
An integral domain is a commutative ring without zero divisors.
The characteristic of a ring is the smallest integer (n) such that (n \cdot a = 0) for all (a) in the ring, or it is 0 if no such (n) exists.
A unit element inside a ring means it has a multiplicative inverse.

Factorization and Irreducibility

In a commutative ring (R), an element (b) is a factor of (a) if there exists a (c) in (R) such that (b = ac).
An irreducible element in (R) does not factor into the product of two non-unit elements.
Elements understood as associates have the same factors.

Field Extensions

A field (K) is an extension of a field (F) if (F) is a subfield of (K).
Degree of a field extension (K/F) represents the dimension of (K) as a vector space over (F) and is denoted as ([K:F]).

Important Relationships Among Classes of Rings

The chain of properties indicating relationships among ring classes is as follows:
- Every Euclidean Domain (ED) is a Principal Ideal Domain (PID).
- Every PID is a Unique Factorization Domain (UFD).
- A UFD is also an Integral Domain (ID).

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Description

Explore the properties of ideals in ring theory with this quiz. Understand the mapping between ideals and their intersections. Test your knowledge on key concepts and definitions within this mathematical framework.