Research Journal of Mathematics and Statistics 1(1): 1-3, 2009
ISSN: 2040-7505
© M axwell Scientific Organaization 2009
Submitted Date: June 01, 2009
Accepted Date: June 10, 2009
Published Date: September 23, 2009
On the Fundamental Theorems of ' - Regular Ring
1
D. Krishnaswamy and 2 N. Kum aresan
Annam alai University, A nnamalai nag ar, Ch idam baram. India
2
CA S in M arine Biology , Annam alai University, Parangipettai,
1
Abstract: In this paper, we have defined ' - Regular ring and characterization of '- Regular ring have been
established analogous to Barnes ' - ring and Von – Neumann regular ring. Furthe r we have defined ideal in
' - Regular ring and fundamental theorems have been determined analogous to the fundamental theorem on
rings found in He rstein.
Key w ords: Homom orphism, ideals, isom orphism, ' - ring, regular ring, '- Regular ring.
INTRODUCTION
Proof: If a – aya is
- regular, then there exists
y,z0 such that (a – aya) z (a – aya) = (a – aya) shows
that axa =a. Thus, a is
- regular.
An element a of a ring R is said to be regular if and
only if there exist an elemen t x of R such that axa = a.
The ring R is regular if and only if each elemen t of R is
regular. The concept of regular ring was introduced by
Von Newm ann (1936). A ccording to B arnes (196 6) ' ring have been defined in the following way. Let R and '
be two additive abelian groups. If for all x, y, z 0 R and for
all, ",$ , ' the conditions
Definition: 1 .3: Let (R,
) be a ring pair. Let
the set of all mxn matrices over the
- ring R. Let
be the set of all nxm matrices over the R – ring
ring
is said to be
- regular ring if for
there exists
(1)
be
. A
,
such that AXA = A.
Let R n denotes the square matrix of order n over the
- ring R.
(2)
(3)
Theorem : 1.1: Let R,
be a ring pair. R is a
regular ring if and on ly if R n is a
- regular ring.
are satisfied, and then R is called a
- ring. Th at is R is
a ring with respect to '. Authors like Nob usaw a (1964),
Ravishankar et al, (1979) and Barnes (1966) himself have
extended many fundamental results in ring theory to
- ring. Further Hatice Kandamar (2000) has studied Kderivation of a Gamma Ring. Here we have developed
regular ring,
- ring to
- regular ring and fundamental
theorems on
- regular ring have been established.
-
Proof: If R is a
- regular, then for
, there is an
element
such that xax = x. Now, let
Definition: 1.1: Let R and
be two additive abelian
groups. An element a , R is said to be
- regular if
and only if there exists an element x0 such that
axa = a. A
- ring is said to be a
- regular if and only
if each eleme nt of R is
- regular
.
Definition: 1.2: If R is a
- ring and
is a R – ring,
then (R, ) is called a ring pair. Throughout, we consider
(R, ) to be a ring pair.
A=
and X =
Y=
Z=
and denote A – AXA = B, BYB – B =
C and C – CZC = 0. Thus, C is a
- regular in R 2 .
Hence by lemma 1.1, B is a
- regular in R 2 . Again
applying by lemma 1.1, A is a
- regular in R 2 .
If A ,R n let A 1 be the ma trix of
with A in the
upper left hand corn er and zero’s elsew here. N ow, as
element of
A 1 is
- regular, then there exists a
element
X=
Such that A ! X A 1 = A.
A - AX A =0 an d hence A is
- regular.
Conversely, if R n is
- regular, then it is clear that R is
- regular.
Lemm a: 1.1: Let (R, ) be a ring pair. For a0
and
y0 such that a – aya is
- regular, then a is
- regular.
Corresponding Author: N. Kumaresan, CAS in Marine Biology, Annamalai University, Parangipettai, India
1
Res. J. Math. Stat., 1(1): 1-3, 2009
Theorem: 1.2: Let (R,
regular if and only if is a
Proof: By theorem 1.1
is a
) be a ring pair. R is a
- regular ring.
, is
is a
- regular ring. We shall prove R/K .
Consider the mapping: R/K – Ê
Consider the mapping
Ê such that
-
- regular ring. Since
- regular ring for all
.
, there ex ists
such that B X B = B. For
To prove:
there
such that A Y A = A by defining
exists
and A =
, it follows that
(i) n is well defined and one to one.
(ii) n[(k + a"c) + (k + b"c)] = n (k + a"c) + (k + b"c)
(iii) n(x + a"c) "(y + a"c) = n(x + a"c)n (")n(y +a"c),
. Since A is arbitrary in
is
it will be proved by usual way of Fundamental theorem
on homomorphism of groups. Hence n is isomorphic from
R/K
Ê.
- regular ring.
Similarly, we have to prove the converse part also by
using theorem 1.1.
Theorem: 2.2: Let R and
be
- regular rings. and
n be a homomorphism from R onto Ê with kernel k.
Then, R/K
. Moreover there exist one to one
correspondences between the set of ideals of and the set
of ideals
which contains the same K.
This
correspondence can be achieved by associating with an
ideal w ith
in
to an ideal W defined by
W ={
} in R. Then , R/W
Ideals in - regular ring:
Definition: 2.1: A subset S of the
- regular ring R is a
right ideal (left ideal) of R, if S is a subgroup of R and
S
R=
is contained in S.
A subset S in a
- regular ring R is a two sided ideal
or an ideal in
- regular ring R if it contain both left and
right ideals.
(or).
Proof: Consider the mapping
is
Definition: 2.2: If A and B are ideals in
- regular ring
R, then the sum o f A and B is also an ideal of R. i.e.,
A + B = {a + b: a,A,b,B}.
defined by
Definition: 2.3: Let A be an ideal in ' - regular ring R,
then the set R/A is defined by R/A = {x +
a
} under the operation (+, C) is
(i) R is well defined and one to one.
(ii) R(a 1 "c +a2 "c) = R(a 1 "c) + R(a 2 "c) and
(iii) R[(a 1 "c) " (a 2 "c)] = R(a 1 "c) R(") R(a 2 "c)
To prove:
defined as follows
i)
ii)
(x + a
(x + a
c) + (y + a
c) " (y + a
Then, (R/A , +, C ) form a
it will be proved by usu al way o f Fundamental theorem
on homomorphism of groups. The kern el of
homomorphism is defined by
c) = (x + y) + a c
c) = (x
y) + a c.
Ker R = {
- regular ring R.
}
Now
Definition: 2.4: Let R and R3 be
- regular rings. Let
n be a mapping from R v R3 , then if ( a"c) = bVc.
If n is a homomorp hism of R v Ê , then the kernel of
homom orphism of n is defined by Ker n = {r 0 R:
n(r) = ó} where 0 is the zero ideal of R and ó is the zero
ideal of Ê.
is a homomorphism with kernel K.
Thus,
Fundamental
Theorem
on homom orphism in
'- regular rings
Theorem: 2.1: Every homomorphic image of a
regular ring R is isomorphic to some residue class ring
analogo us to He rstein (1975).
By using Theorem 2.1, w e get
.
By restricting the homomorphism between the ideals W of
R and
of havin g the same kerne l K an d aga in
by applying Theorem 2.1, We obtain
Proof: Let Ê be the hom omo rphic image of a
- regular
ring R and be the corresponding homomorphism. Then,
f is a homo morph ism of R onto Ê. Let K be the kernel of
this homomorphism. Then K is an ideal of R. Then , R/K
.
2
and hence
Res. J. Math. Stat., 1(1): 1-3, 2009
Nobusawa, N., 19 64. O n a generalization of the ring
theory, Osaka J. Math., 1: 81-89.
Ravi Shankar, T.S. and U.S. Sukla, 1979. Structure of
Gamma rings, Pacific J. Math., 80 (2): 537-559.
Von Neumann, J., 1936. On regular rings Proc. National
Academic Science U.S.A., 22: 707-713.
REFERENCE
Barnes, W.E., 1966. On the -Ring of the Nobusawa,
Pacific J. M ath., 18(3): 411 -422.
Kan dama r, H., 2000 . The k-derivation of a Gamma rings,
Turk J. Math., 24: 221-231
Herstein, I.N., 1975. Topics in Algebra, 2 n d Edn., Prentice
Hall of India, New Delhi.
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