Introduction to Rings
Subject: ALGEBRA-III
Semester-IV
Lesson: Introduction to Rings
Lesson Developer: Divya Bhambri
College/Department: St. Stephen’s College,
University of Delhi
Institute of Life Long Learning, University of Delhi Page 1
, Introduction to Rings
Contents
1. INTRODUCTION ............................................................................ 3
2. RINGS ......................................................................................... 3
2.1. Definition and Examples of Rings ............................................... 3
Properties of Rings ............................................................................. 7
2.2. Special types of Rings .............................................................. 8
Boolean Ring .................................................................................. 8
Division Ring ................................................................................... 9
The Ring of Quaternions ................................................................. 10
Field ............................................................................................ 11
3. SUBRINGS ................................................................................. 12
4. Integral Domain ......................................................................... 15
5. Nilpotent and Idempotent elements .............................................. 18
6. Characteristic of a Ring ................................................................ 20
Exercises ........................................................................................... 22
References ........................................................................................ 23
Suggested Readings ........................................................................... 23
Institute of Life Long Learning, University of Delhi Page 2
, Introduction to Rings
1. INTRODUCTION
A ring is an algebraic structure with two binary operations in which arithmetic
operations of addition and multiplication on the set of integers are generalized so
that results from arithmetic are extended to non-numerical objects
like polynomials, matrices and functions.
In 1892, David Hilbert introduced the term "Zahlring" which means number
ring and it was formally published in 1897. The word "Ring" could mean
"association". According to Harvey Cohn, the term ring was used by D. Hilbert that
had the property of "circling its elements directly back to elements of itself”. In
particular, for ring of algebraic integers, all high powers of an algebraic integer can
be written as an integral combination of a known set of lower powers, and thus the
powers "cycle back". For example, if then
and so on; in general, we observe
that is an integral linear combination of and .
2. RINGS
The modern axiomatic definition of (commutative) rings was given by Emmy
Noether in 1920, which led to the development of commutative ring theory.
2.1. Definition and Examples of Rings
Definition: Let be a non-empty set with two binary operations addition and
multiplication (denoted by + and .). If the following axioms are satisfied for all
:
1.
2.
3. –
4.
5.
6.
Then we say that is a ring or we can simply write it as is a ring, in case
there is no scope of confusion about the operations + and . .
If along with these six conditions, elements of also satisfy:
7.
Then we say that is a commutative ring.
Institute of Life Long Learning, University of Delhi Page 3
Subject: ALGEBRA-III
Semester-IV
Lesson: Introduction to Rings
Lesson Developer: Divya Bhambri
College/Department: St. Stephen’s College,
University of Delhi
Institute of Life Long Learning, University of Delhi Page 1
, Introduction to Rings
Contents
1. INTRODUCTION ............................................................................ 3
2. RINGS ......................................................................................... 3
2.1. Definition and Examples of Rings ............................................... 3
Properties of Rings ............................................................................. 7
2.2. Special types of Rings .............................................................. 8
Boolean Ring .................................................................................. 8
Division Ring ................................................................................... 9
The Ring of Quaternions ................................................................. 10
Field ............................................................................................ 11
3. SUBRINGS ................................................................................. 12
4. Integral Domain ......................................................................... 15
5. Nilpotent and Idempotent elements .............................................. 18
6. Characteristic of a Ring ................................................................ 20
Exercises ........................................................................................... 22
References ........................................................................................ 23
Suggested Readings ........................................................................... 23
Institute of Life Long Learning, University of Delhi Page 2
, Introduction to Rings
1. INTRODUCTION
A ring is an algebraic structure with two binary operations in which arithmetic
operations of addition and multiplication on the set of integers are generalized so
that results from arithmetic are extended to non-numerical objects
like polynomials, matrices and functions.
In 1892, David Hilbert introduced the term "Zahlring" which means number
ring and it was formally published in 1897. The word "Ring" could mean
"association". According to Harvey Cohn, the term ring was used by D. Hilbert that
had the property of "circling its elements directly back to elements of itself”. In
particular, for ring of algebraic integers, all high powers of an algebraic integer can
be written as an integral combination of a known set of lower powers, and thus the
powers "cycle back". For example, if then
and so on; in general, we observe
that is an integral linear combination of and .
2. RINGS
The modern axiomatic definition of (commutative) rings was given by Emmy
Noether in 1920, which led to the development of commutative ring theory.
2.1. Definition and Examples of Rings
Definition: Let be a non-empty set with two binary operations addition and
multiplication (denoted by + and .). If the following axioms are satisfied for all
:
1.
2.
3. –
4.
5.
6.
Then we say that is a ring or we can simply write it as is a ring, in case
there is no scope of confusion about the operations + and . .
If along with these six conditions, elements of also satisfy:
7.
Then we say that is a commutative ring.
Institute of Life Long Learning, University of Delhi Page 3
