A set with addition and multiplication such that addition forms an abelian group, multiplication is associative and has an identity, and multiplication distributes over addition.

What does it mean for addition to form an abelian group?

Addition is associative and commutative, has an additive identity, and every element has an additive inverse.

What is a multiplicative identity?

An element $1$ satisfying $a1=1a=a$ for every ring element $a$.

Must multiplication in a ring be commutative?

No. A general ring only requires associative multiplication; a ring whose multiplication is commutative is called a commutative ring.

How are operations performed in $\mathbb{Z}/4\mathbb{Z}$?

Add or multiply representatives as integers, then replace the result by its remainder modulo $4$.

Abstract algebraLesson 1 of 5 in the course Rings and ideals

Ring Theory: Axioms, Ideals, Quotients, and Classifications

5 flashcards · 4 quiz questions · 5 match pairs · 5 fill-in sentences · English

Save course

Lesson progress

0 / 5

Your progress is kept in this browser only.Sign up to save it

Summary

A ring is an algebraic structure with addition and multiplication that generalizes familiar arithmetic while allowing multiplication to be noncommutative. This lesson introduces the ring axioms and the role of the additive and multiplicative identities.

A ring $R$ is a set equipped with two binary operations, usually written $+$ and multiplication. A binary operation takes two elements of the set and returns another element of the set. Thus, for $a,b\in R$ , both $a+b$ and $ab$ must belong to $R$ .

All flashcards

What is a ring?: A set with addition and multiplication such that addition forms an abelian group, multiplication is associative and has an identity, and multiplication distributes over addition.
What does it mean for addition to form an abelian group?: Addition is associative and commutative, has an additive identity, and every element has an additive inverse.
What is a multiplicative identity?: An element $1$ satisfying $a1=1a=a$ for every ring element $a$ .
Must multiplication in a ring be commutative?: No. A general ring only requires associative multiplication; a ring whose multiplication is commutative is called a commutative ring.
How are operations performed in $\mathbb{Z}/4\mathbb{Z}$ ?: Add or multiply representatives as integers, then replace the result by its remainder modulo $4$ .

Quiz questions

What algebraic structure do the elements of a ring form under addition?Answer: An abelian group.
What follows from the ring axioms about multiplication by zero?Answer: Both $a0=0$ and $0a=0$ for every $a$ .
If $I$ is a two-sided ideal of $R$ , what are the elements of the quotient ring $R/I$ ?Answer: Additive cosets $a+I$ of the ideal $I$ .
What is the product $\overline{3}\,\overline{3}$ in $\mathbb{Z}/4\mathbb{Z}$ ?Answer: $\overline{1}$ .

Based on the Wikipedia article Ring (mathematics), licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

All Abstract algebra lessons

Abstract algebraIntegral Domains: Zero Divisors, Divisibility, and Fraction Fields5 flashcards · 5 quiz questions · 4 gamesPlay free →Abstract algebraGroup Actions: Axioms, Orbits, Counting, and Representations5 flashcards · 5 quiz questions · 4 gamesPlay free →Abstract algebraPolynomial Rings: Evaluation, Factorization, Quotients, and Multivariate Structure5 flashcards · 5 quiz questions · 4 gamesPlay free →Abstract algebraCyclic Groups: Generators, Orders, and Structural Properties5 flashcards · 5 quiz questions · 4 gamesPlay free →Abstract algebraGroup Theory: Axioms, Symmetries, and Structural Ideas5 flashcards · 5 quiz questions · 4 gamesPlay free →Abstract algebraSubgroups: Tests, Generation, Cosets, and Lagrange's Theorem5 flashcards · 5 quiz questions · 4 gamesPlay free →

Summary

All flashcards

Quiz questions

More in Abstract algebra