3. Ring:
A Non Empty set R is called ring if,
❖(R,+) is abelian group.
❖(R, .) is semigroup.
❖Distributive law hold.
4.
5.
6. Let (R,+, .) be a ring and S be non empty subset of R, then S is said to be a
subring of R if S itself a ring under same binary operation in R.
NOTE:
Every ring R has atleast two subrings
i.e.
R itself and {0}.
and these subrings are called trivial subrings.
7.
8. Ideals were first proposed by Richard Dedekind in 1876 in the third
edition of his book (English: Lectures
on NumberTheory).They were a generalization of the concept of ideal
numbers developed by Ernst Kummer. Later the concept was
expanded by David Hilbert and especially Emmy Noether.
Richard Dedekind
9. Ideals:
ra ε S for all aεS and rεR.
ar ε S for all aεS and rεR.
10. Z={0,±1,±2,±3,±4,±…………}
2Z={0,±2,±4,±6,±8,±………..}
Clearly set of even integers is ideal of ring Z.
NOTE:
• Z and Q be ring and Z be subring of rational number Q, but Z is not an ideal of
ring Q.
• Real number R be a ring and Q be a subring of R, but Q is not ideal of ring R.
11. A ring which has no proper ideal.
OR
IF {0} and {R} be only ideals of ring R then R is called simple ring.
NOTE:
Every ring is an ideal of itself.
If R is a ring then {0} and {R} be the only
ideals of ring R.
12. Let R be a commutative subring with unity and let I be an ideal of R, then I become principal ideal if element
in ideal I generated by a single element.
i.e.
I= <a>
Example:
▪ I={0,±2,±4,±6,±8±……..}
I= <2n>
▪ 3 is also generator if 3z i.e. 3z=<3>
13. An ideal M≠R is called maximal ideal if there does not exist any proper ideal of
R containing M.
let 𝑰 𝟏, 𝑰 𝟐, 𝑰 𝟑 be ideal of ring R, if 𝑰 𝟏 ⊂ 𝑰 𝟐, and 𝑰 𝟑 ⊂ 𝑰 𝟐 then I2 is an maximal
ideal. -
Example:
z= set of integers
2z is maximal ideal of z because 2z has no proper ideal of z containing 2z.
14. Let R be a ring and P be an ideal of R than an ideal P≠R is prime ideal if
a, bεR then
a.b εP either a ε P or b ε P
EXAMPLE:
Z={0,±1,±2,±3±……..}
P=2Z={0,±2,±4,±6±……..}
p=2z is an prime ideal of R=Z.
16. ❖ Union OfTwo Ideals:
Union of two ideals may or may not be an ideals.
EXAMPLE:
𝒁 = 𝟎, ±𝟏, ±𝟐, ±𝟑, … & 𝟑𝒁 = 𝟎, ±𝟑, ±𝟔, ±𝟗, …
Then 𝟐𝒁 ∪ 𝟑𝒁 = {𝟎, ±𝟐, ±𝟑, ±𝟒, ±𝟔, … }
𝟐 , 𝟑 ∈ 𝟐𝒁 ∪ 𝟑𝒁
For subring….. 𝟐 − 𝟑 = −𝟏 does not belong to subring… so union
of two ideal is not an ideal of Z.
1. May Not be Ideal
17. ❖ Union OfTwo Ideals:
Example:
As 𝒁 = 𝟎, ±𝟏, ±𝟐, ±𝟑, … & 𝟐𝒁 = 𝟎, ±𝟐, ±𝟒, ±𝟔, …
Also 𝟒𝒁 = {𝟎, ±𝟒, ±𝟖, ±𝟏𝟐, … }
Then 𝟐𝒁 ∪ 𝟒𝒁 = {𝟎, ±𝟐, ±𝟒, ±𝟔, … }
As 𝟐𝒁 ∪ 𝟒𝒁 is an even integer, As even Integer is an Ideal of Z
So Union of two Ideals may be Ideals.
2. May be Ideals