Questions and Answers
What is a ring?
What defines a commutative ring?
A ring with commutative multiplication.
What is a unit of a ring?
A non-zero element of a commutative ring with unity that has a multiplicative inverse.
What is the definition of a direct sum of rings?
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What theorem states rules of multiplication in a ring?
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A non-zero element of a commutative ring that can yield zero when multiplied with another non-zero element is called a zero divisor.
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What is the characteristic of a ring?
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What is a prime ideal?
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What is an integral domain?
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A proper ideal of R is called a ______.
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Study Notes
Ring
- A ring ( R ) is a non-empty set equipped with two binary operations: addition and multiplication.
- Essential properties include:
- Commutativity of addition
- Associativity of addition and multiplication
- Existence of additive identity (0)
- Existence of additive inverses
- Associativity of multiplication
- Distributive property linking multiplication and addition.
Commutative Ring
- A ring where multiplication is commutative, meaning ( ab = ba ) for all ( a, b \in R ).
Ring with Unity
- A ring that contains a nonzero multiplicative identity (often denoted as 1).
Unit of a Ring
- An element in a commutative ring with unity that possesses a multiplicative inverse.
Direct Sum of Rings
- Denoted ( R_1 \oplus R_2 \oplus ... \oplus R_n ), consisting of tuples from each ring with componentwise addition and multiplication.
Theorem: Rules of Multiplication in a Ring
- Fundamental properties include:
- Any element multiplied by zero results in zero.
- The product of an element and the negation is the negation of the product.
- Products of negatives yield a positive.
- Distributive properties of multiplication over subtraction.
Theorem: Uniqueness of the Unity and Inverse
- If a ring has a unity, it is unique; similarly, the multiplicative inverse for any element is also unique.
Subring
- A subset ( S ) of a ring ( R ) that is also a ring itself under the operations of ( R ).
Theorem: Subring Test
- A non-empty subset ( S ) of a ring is a subring if it is closed under subtraction and multiplication.
Zero Divisor
- A non-zero element ( a ) of a commutative ring such that there exists a non-zero element ( b ) with ( ab = 0 ).
Integral Domain
- A commutative ring with unity containing no zero divisors, ensuring cancellation property holds.
Theorem: Cancellation Property
- In an integral domain, if ( a \neq 0 ) and ( ab = ac ), then ( b = c ).
Field
- A commutative ring with unity where every nonzero element is a unit, forming a group under multiplication.
Theorem: Finite Integral Domains are Fields
- Any finite integral domain qualifies as a field.
Characteristic of a Ring
- Defined as the smallest positive integer ( n ) such that ( nx = 0 ) for all ( x \in R ); if nonexistent, the characteristic is 0.
Theorem: Characteristic of a Ring with Unity
- In a ring with unity, if unity has infinite order under addition, the characteristic is 0; otherwise, it is equal to ( n ) if order exists.
Theorem: Characteristic of an Integral Domain
- The characteristic of an integral domain can only be 0 or a prime number.
(Two-Sided) Ideal
- A subring ( I ) of a ring ( R ) that absorbs multiplication from ( R ), satisfying closure under subtraction.
Proper Ideal
- An ideal ( I ) that forms a proper subset of ring ( R ).
Theorem: Ideal Test
- A non-empty subset ( I ) is an ideal of ( R ) if it is closed under subtraction and absorbs multiplication from ( R ).
Principal Ideal Generated by ( a )
- The set ( { ra | r \in R } ) represents a principal ideal generated by element ( a ) in a commutative ring with unity.
Ideal Generated by Elements ( a_1, a_2, ..., a_n )
- A set ( \langle a_1, a_2, ..., a_n \rangle = { r_1a_1 + r_2a_2 + ... + r_na_n | r_i \in R } ) characterizes the ideal generated by elements.
Theorem: Existence of Factor Rings
- For a ring ( R ) and a subring ( A ), the set of cosets ( { r + A | r \in A } ) forms a ring if ( A ) is an ideal of ( R ).
Prime Ideal
- A proper ideal ( I ) such that if ( ab \in I ) for ( a, b \in R ), then either ( a \in I ) or ( b \in I ).
Maximal Ideal
- A proper ideal ( I ) such that within any ideal ( B ) containing ( I ), it holds that ( B = I ) or ( B = R ).
Theorem: ( R/I ) is an Integral Domain if and Only if ( I ) is Prime
- For a commutative ring with unity ( R ) and an ideal ( I ), the quotient ( R/I ) is an integral domain if ( I ) is prime.
Theorem: ( R/I ) is a Field if and Only if ( I ) is Maximal
- For a commutative ring with unity ( R ) and an ideal ( I ), the quotient ( R/I ) is a field if ( I ) is maximal.
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Test your knowledge on the essential concepts of rings in abstract algebra with these flashcards. Each card presents a key term along with its definition, helping you grasp fundamental ideas like commutative rings and more. Perfect for students seeking to reinforce their understanding of algebraic structures.
