Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

UNIT-IV
77
A. B ≠ B. A.
If IR, the set of real numbers is replaced by E, the set of even integers, then
(M 2
(E), + ;) is a non-commutative ring without identity, as
4. Z 4
: Set of integers module 4.
is a commutative ring with identity
, where +, ; are defined shown in following tables:
In bZ
bbg
= m
g r r
Important Remark:
. 4 0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 0 2
3 0 3 2 1
As we saw in a group that Cancellation law holds but in a ring the cancellation law may fail for multiplication:
g
6
, + , . , 2. 3 = 0 = 4. 3 but 2 ≠4.
∉ E. F B A
,
H G I K J Definition F H G I K J Subring:
b 0 0 1 F =
H G
0 0 I φZ ≠+ o
K J
4 :, S, 4 + ∈, 1R ;
2 1 , b3
∈,
R3
∀ a,
b ∈R
1 a 4∃
u ~ −
S . ∈
b
1
= 0∈L
R
= 0R
−
ab0, = 2, , a4 4. 1 20 = 2b, 4=
. 04 = 4.
0 0 112 0 Let (R,
3 0 0+, .) is a ring 1 0and
a non-empty subset of R. Then bS, + , . g(with same binary operations) is called
a subring if
1 1 2 3 0
1. Closure
2 2 3 0 1
3 3 0 1 2
2. ∀ a ∈ R, − a∈R.
Examples:
(1) bZ 6
, + ,. gis a commutative ring with identity.
(S, +, .) is a subring with identity (multiplicative)
, since
Note that parent ring Z 6 , + ; ghas identity . This shows that a
subring may have a different identity from that of a given ring.
Definitions: Units in a ring
Let R be a commutative righ with identity 1. An element a
that a.b = l. The element a
Divisiors of Zero
If and ab = 0 for some non zero
universe of a (if it exists)
∈ R is called a Unit of R.
b
∈ R is said to be invertible if
such
. Then ‘a’ cannot be unit in R, since multiplying ab = 0, by the