Lecture on Gamma Nearrings

2009 DecemberDept Maths, Periyar Maniammai University, Tanjavur

Lecture by Prof. Dr Bhavanari Satyanarayana, Acharya Nagarjuna University, A.P. :1:

Fuzzy Ideals in -Near-rings

Prof. Bhavanari Satyanarayana

AP SCIENTIST-2009 Awardee

Department of Mathematics

Acharya Nagarjuana University

Nagarjuna Nagar 522 510

Andhra Pradesh, India.

The concept -ring, a generalization of a ring was introduced by Nobusawa [1]

and generalized by Barnes [1]. A generalization of both the concepts near-ring and the

gamma-ring, namely -near-ring was introduced by Satyanarayana [1] and later studied

by several authors like Booth [ 1, 2, 3 ], Booth & Greonewald [ 1, 2, 3 ], Jun ,

Sapanci , & Ozturk [ 1 ], Satyanarayana [ 1, 2, 3, 4 ], Satyanarayana & Syam

Prasad [ 1 ], Selvaraj & George [ 1, 2 ] , Syam Prasad [ 1 ], Syam Prasad &

Satyanarayana [ 1 ], Mustafa, & Mehmet Ali [ 1 ]

1. Fundamental Definitions & Results

1.1 Definition (Satyanarayana [1]): Let (M, +) be a group (not necessarily

Abelian) and be a non-empty set. Then M is said to be a -near-ring if there

exists a mapping M M M (the image of (a, , b) is denoted by ab),

satisfying the following conditions:

(i) (a + b)c = ac + bc; and

(ii) (ab)c = a(bc) for all a, b, c M and , .

M is said to be a zero-symmetric -near-ring if ao = o for all a M and

, where o is the additive identity in M.

A natural Example of -near-ring is given below:

1.2 Example (Satyanarayana [1]): Let (G, +) be a group and X be a non-empty

set. Let M = {f / f:X G}. Then M is a group under point wise addition.

If G is non-abelian, then (M, +) is non - abelian. To see this, let a, b

such that a + b b + a. Now define fa, fb from X to G by fa(x) = a and fb(x) = b

for all x X.

It is clear that fa, fb M and fa + fb fb + fa.

Thus if, G is non-abelian, then M is also non-abelian.

Let be the set of all mappings of G into X.

If f1, f2 M and g , then, obviously, f1gf2 M.

For all f1, f2, f3 M and g1, g2 , it is clear that

i) (f1gf2)g2f3 = f1g1(f2g2f3); and

ii) (f1+f2)g1f3 = f1g1f3 + f2g1f3.

But f1g1(f2 +f3) need not be equal to f1g1f2 + f1g1f3.

To see this, fix o z G and u X.

Define Gu: G X by gu(x) = u for all x G and

fz:X G by fz(x) = z for all x X.

Now for any two elements f2, f3 M, consider

fzgu(f2+ f3) and fzguf2 + fzguf3. For all x X

[fzgu(f2+ f3)] (x) = fz[gu(f2(x) + f3(x))] = fz(u) = z and

[fzguf2 + fzguf3](x) = fzguf2(x) + fzguf3(x) = fz(u) + fz(u) = z + z.

Since z o, we have z z + z and hence fzgu(f2+ f3) fzguf2 + fzguf3.

Now we have the following:

If (G, +) is non-abelian and X is a non-empty set then M = { f / f: X G } is a

non-abelian group under pointwise addition and there exists a mapping

Where = { g / g: G X } satisfying the following conditions:

i) (f1gf2)g2f3 = f1g1(f2g2f3); and

ii) (f1+f2)g1f3 = f1g1f3 + f2g1f3

for all f1, f2,f3 M and for all g1, g2 . Therefore M is a -near-ring.

1.3 Definition (Satyanarayana [1]): Let M be a -near-ring. Then a normal

subgroup I of (M, +) is called

(i) a left ideal if a(b + i) - ab I for all a, b M, and i I;

(ii) a right ideal if ia I for all a M, , i I; and

(iii) an ideal if it is both a left and a right ideal.

1.4 Definition: (Satyanarayana [1]): An ideal A of M is said to be prime if B

and C are ideals of M such that BC A implies B A or C A.

1.5 Definition (Satyanarayana [1]): Let M1 and M2 be -near-rings. A group

homomorphism f of (M1, +) into (M2, +) is said to be a -homomorphism if

f(xy) = f(x)f(y) for all x, y M and .

We say that f is a -isomorphism f is one-one and onto.

For an ideal I of a -near-ring, the quotient -near-ring M/I defined as usual.

1.6 Theorem (Satyanarayana [1]): Let I be an ideal of M and f, the canonical

group epimorphism of M onto M/I. Then f is a -homomorphism of M onto M/I

with kernal I. Conversely if f is a -epimorphism of M1 onto M2 and I is the kernal

of f then M1/I is isomorphic to M2.

1.7 Theorem (Satyanarayana [1]): Let f be a -homomorphism of M1 onto M2

with Kernal I and J*, a non-empty subset of M2. Then J* is an ideal of M2 if and

only if f-1(J*) = J is an ideal of M1 containing I. In this case we have M1/J, M2/J* and

(M1/I)/ (J/I) are -isomorphic.

1.8 Example (Satyanarayana [1]): Let G be non-trivial group and X be a non-

empty set. If M is the set of all mappings from X into G and be the set of all

mappings from G into X, then M is a -near-ring. Let y be a non-zero fixed

element of G. Define : X G by (x) = y for every x X. Then o M, where

o is the additive identity in M and go = o for any g . Therefore M is a -

near-ring, which is not zero symmetric.

1.9 Notation: For any two subsets A, B of M the set { ab | aA, , bB} is

denoted by either AB or AB. {xA| xB} is denoted by A \ B. For any subset X

of M, the smallest ideal containing X is denoted by <X>. If X = {a} then <X> is

denoted by <a>.

2. The f-Prime Radical in -Nearrings

Satyanarayana [2] introduced the concepts of f-prime ideal and f-prime radical in

-near-rings, and obtained a characterization of f-prime radical in terms of f-

strongly nilpotent elements.

Throughout this section f stands for a mapping from M into the set of all

ideals of M, satisfying the following conditions:

(i) a f(a);

(ii) x f(a) + A, A is an ideal f(x) f(a) + A

Such type of mappings may be called as ideal mappings.

A natural example for this is given here. Let M be a -near-ring and Q M.

Define, for each a M, f(a) = <{a} U Q>, the ideal generated by the union of Q

and {a}.

Then f satisfies the above two conditions, and hence f is an ideal mapping.

2.1 Definition (Satyanarayana [2]): A subset H of M is said to be

(i). an m-system if, for every h1, h2 H there exist

h11 <h1 > and h1

2 < h2>, such that h11h1

(ii). an f-system if H contains an m-system H*, called a kernal of H, such that, for

every h H, f(h) ∩ H* . In this case we write that H(H*) is an f-system.

2.2 Definition (Satyanarayana [2]): An ideal A of M is said to be

(i) Prime if B and C are ideals of M such that BC A B A or C A.

(ii) f-prime if M\A is an f-system.

2.3 Note (Satyanarayana [2]) The following statements are clear.

(i) A is a prime ideal if and only if M\A is an m-system;

(ii) Every m-system is an f-system.

(iii) A is a prime ideal M\A is an m-system M\A is an f-system A is f-prime

(iv) Every f-prime ideal need not be a prime ideal.

2.4 Example (Satyanarayana [2]): Let N1 be a near-ring with a non-nilpotent

element x. Let N2, N3 be near-rings.

Consider M = N1 N2 N3 , the near ring which is the direct sum of N1, N2, N3.

Write = {.}, where “.” is the product in M.

Now, M is a -near-ring and Ii = Ni, 1 i 3 are ideals of M.

Write S* = {x, x2, x3, ...} and f(a) = <{a, x}> for all a M.

Now S* is an m-system, S* M\I2 and M\I2 is an f-system with kernal S*.

Therefore I2 is an f-prime ideal.

But I2 is not a prime ideal because I1 I2, I3 I2 and I1 I3 I2 .

Hence, in general, every f-prime ideal need not be a prime ideal.

2.5 Definitions (Satyanarayana [2]): (i) A subset H of M is said to be nilpotent

if Hn = {0} (that is, H H...H = {0} for some integer n 2.

(ii) An element a M is said to be nilpotent if {a}n = 0, that is, (a )n-1a = {0}

for some n 2.

(iii) A subset H of M is said to be nil if every element of H is nilpotent.

(iv) An element a M is said to be f-nilpotent (resp. f-nil) if f(a) is nilpotent

(resp. nil).

(v) A subset H of M is said to be f-nil if every element of H is f-nilpotent.

2.6 Remark (Satyanarayana [2]) Let a M and H M. Then the following holds:

(i) a is f-nilpotent a is f-nil a is nilpotent;

(ii) H is f-nilpotent H is f-nil H is nil;

(iii) H is f-nilpotent H is nilpotent H is nil.

2.7 Examples (Satyanarayana [2]) (i) Let N be a near-ring with x, y N such

that x is nilpotent and y is not nilpotent.

Now, M = N is a -near-ring with = {.}.

Write f(a) = <{a, y}> for all a M.

Now, y f(a), y is not nilpotent and so f(a) is not nil.

So x is not f-nil but it is nilpotent.

(ii) If Q is an ideal of N which is nil but not nilpotent, then define

f(a) = <({a}U Q)> for all a M.

For any q Q, we have f(q) = Q and so Q is f-nil but not f-nilpotent.

2.8 Lemma (Satyanarayana [2]) Let P be an ideal of M.

Then, for any two subsets A and B of M, we have (A +P)(B + P) AB + P.

2.9 Lemma (Satyanarayana [2]) Let S (S*) be an f-system in M and let A be an

ideal in M which does not meet S. Then A is contained in a maximal ideal P

which does not meet S. The ideal P necessarily be a f-prime ideal.

2.10 Definition (Satyanarayana [2]) The f-radical (denoted by f-rad (A)) of an

ideal A is defined to be the set of all elements a of M with the property that every

f-system which contains ‘a’ contains an element of A.

2.11 Theorem (Satyanarayana [2]) The f-radical of an ideal A is the intersection

of all f-prime ideals containing A.

2.12 Definition (Satyanarayana [2]) Let A be an ideal of M.

An element a in M is said to be strongly nilpotent modulo A if, for every

sequence x1, x2 , . . . of elements of M such that x1 = a and xi = xi-11 i-1 x*

<xi-1>, there exists an integer k such that xs A for s k.

An element a M is said to be strongly nilpotent if it is strongly nilpotent

modulo (0).

An element x M is said to be f-strongly nilpotent modulo A if every

element of f(x) is strongly nilpotent modulo A.

It is clear that every f-strongly nilpotent element is strongly nilpotent. The

following example establishes that the converse is not true.

2.13 Example (Satyanarayana [2]) Let N be a near-ring such that (0) does not

equal the prime radical of N N.

Let x N \ (prime radical of N).

We consider M = N as a -near-ring with = {.}.

Write f(a) = <{a, x}> for every element a N.

Now by a known result (the prime radical of a near-ring N is the set of all strongly

nilpotent elements of N) we get that x is not strongly nilpotent.

Since x f(a) for all a, we have that no element of N is f-strongly nilpotent, where

all elements of the prime radical of N are strongly nilpotent.

2.14 Lemma (Satyanarayana [2]) Let a1 , a2 , . . . be a sequence of elements of

M with ai = a1i-1 i-1a*

i-1, for some i-1 and a1i-1, a*

i-1 <ai-1>.

Then {ai | i 1} is an m-sequence.

2.15 Theorem (Satyanarayana [2])

f-rad M = {x M| x is f-strongly nilpotent} U {0}

2.16 Theorem (Satyanarayana [2]) If A is an ideal of M, then

f-rad (A) = {x M | x is f-strongly nilpotent modulo A} U A.

Some aspects of radical theory (Jocobson radical type, etc) were studied by

Booth [ 1,2,3] and Booth & Gronewald [ 1,2,3].

3. Fuzzyness in -Near-Rings

The concept of Fuzzy ideal of a near-ring was introduced by Abou-Zaid [1]

and later it was studied by Datta & Biswas [1].

Jun, Sapanci and Ozturk [1] intoruced the concept of “fuzzy ideal” in

-near-rings and studied some fundamental properties.

Henceforth, M stands for a zero-symmetric -near-ring.

3.1 Definition: Let : M [0, 1]. Then is said to be a fuzzy ideal of M if it

satisfies the following conditions:

(i) (x + y) min{(x), (y)};

(ii) (-x) = (x);

(iii) (x) = (y + x – y);

(iv) (xy) (x); and

(v) {(x(y + z) – xy} (z) for all x, y, z M and .

3.2 Proposition (Jun, Sapanci & Ozturk [1]): Let be a fuzzy subset of M. Then

the level subsets t = { x M / (x) t }, t im , are ideals of M if and only if

is a fuzzy ideal of M.

3.3 Note (Satyanarayana & Syam Prasad [1]): (i) If is a fuzzy ideal of M then

(x + y) = (y + x) for all x, y M.

(ii) If is fuzzy ideal of M then (o) (x) for all x M.

Verification: (i) Put z = x + y. Now (x + y) = (z) = ( -x + z + x) (since

is a fuzzy ideal) = ( -x + x + y + x) = (y + x).

(ii) Clearly o = ox for all and x M.

This implies (o) = (ox). Consider (o). Now

(o) = {o(o + x) – oo} (x) (since is a fuzzy ideal of M).

Therefore (o) (x) for all x M.

3.4 Lemma (Satyanarayana & Syam Prasad [1]): Let be a fuzzy ideal of M.

If (x – y) = (o) then (x) = (y) for all x, y M.

3.5 Proposition (Th. 2.2 of Syam Prasad & Satyanarayana [1]): Let I be an ideal

of a -near-ring M and t < s in [0,1]. Then the fuzzy subset defined by

(generalized characteristic function)

is a fuzzy ideal of M.

3.6 Definition: Let X and Y be two non-empty sets and f: X Y.

Let and be fuzzy subsets of X and Y respectively.

Then f(), the image of under f, is a fuzzy subset of Y defined by

(f())(y)

and f-1(), the pre-image of under f, is a fuzzy subset of X defined by

(f -1())(x) = (f(x)) for all x X.

3.7 Lemma (Syam Prasad [1]): Let M and M1 be two -near-rings and f: M M1

be a -near-ring homomorphism. If f is surjective and is a fuzzy ideal of M, then

so is f(). If is a fuzzy ideal of M1 then f -1() is a fuzzy ideal in M.

3.8 Proposition (Syam Prasad [1]): Let M and M1 be two -near-rings,

h: M M1 be an -epimorphism and , be fuzzy ideals of M and M1

respectively; then

(i) h(h-1()) = ;

(ii) h-1(h()) ; and

(iii) h-1(h()) = if is constant on ker h.

3.9 Definition: Let and be two fuzzy subsets of M. Then the fuzzy subset

o of M, defined by

(o)(x) = {min ((y), (z))} if x is expressible as a product x = yz

for some

= o, otherwise, for all x, y, z M.

4. Fuzzy Cosets in -Near-rings

4.1 Definition (Def. 2.1 of Satyanarayana & Syam Prasad [1]): Let be a fuzzy

ideal of a -near-ring M and m M. Then a fuzzy subset m + defined by

(m + )(m1) = (m1 – m) for all m1 M, is called a fuzzy coset of the fuzzy ideal

4.2 Proposition (Lemma 2.2 (i) of Satyanarayana & Syam Prasad [1]): If is a

fuzzy ideal of M. Then x + = y + if and only if (x – y) = (o).

4.3 Corollary (Lemma 2.2 (ii) of Satyanarayana & Syam Prasad [1]):

If x + = y + then (x) = (y).

4.4 Proposition (Lemma 2.2 (v) of Satyanarayana & Syam Prasad [1]):

Every fuzzy coset of a fuzzy ideal of M is constant on the ordinary ideal

M = { x M / (x) = (o) }.

4.5 Corollary (Lemma 2.2 (vi) of Satyanarayana & Syam Prasad [1]):

If z M then (x + )(z) = (x).

4.6 Theorem (Th. 2.4 of Satyanarayana & Syam Prasad [1]): Let be a fuzzy

ideal of M. Then the set of fuzzy cosets of is a -near-ring with respect to the

operations defined by

(x + ) + (y + ) = (x + y) + ; and

(x + )(y + ) = xy + for all x, y M and .

4.7 Proposition (Lemma 2.6 of Satyanarayana & Syam Prasad [1]):

Let be a fuzzy ideal of M; the fuzzy subset of M/, is defined by

(x + ) = (x) for all x M, is a fuzzy ideal of M/.

4.8 Theorem (Th. 3.3 of Satyanarayana & Syam Prasad [1]): If is a fuzzy ideal

of M then the map

: M M/ defined by (x) = x + , x M, is a -near-ring homorphism with

kernal M ={ x M / (x) = (o) }.

4.9 Theorem (Th. 3.3 of Satyanarayana & Syam Prasad [1]): The -near-ring M/

is isomorphic to the -near-ring M/M. The isomorphic correspondence is given

by x + ↦ x + M.

4.10 Lemma (Lemma 3.5 of Satyanarayana & Syam Prasad [1]): Let and be

two fuzzy ideals of M such that and (o) = (o).

Then the fuzzy subset of M/ defined by (x + ) = (x) for all x M is a

fuzzy ideal of M/ such that .

4.11 Notation: The fuzzy ideal of M/ is denoted by /.

4.12 Lemma (Lemma 3.7 of Satyanarayana & Syam Prasad [1]): Let be a fuzzy

ideal of M and be a fuzzy ideal of M/ such that . Then the fuzzy subset

of M defined by (x) = (x + ) for all x M is a fuzzy ideal of M such that .

4.13 Theorem (Th. 3.9 of Satyanarayana & Syam Prasad [1]): Let be a fuzzy

ideal of M. There exist an order preserving bijective correspondence between

the set P of all fuzzy ideal of of M such that and (o) = (o) and the set

of all fuzzy ideal of M/ such that .

4.14 Proposition (Th. 3.11 of Satyanarayana & Syam Prasad [1]): Let

h: M M1 be an epimorphism and is a fuzzy ideal of M1 such that = h-1().

Then the map : M/ M1/ defined by (x + ) = h(x) + is a -near-ring

isomorphism.

5. Fuzzy Prime ideals of -near-rings

5.1 Definition (Def. 2.1 of Syam Prasad & Satyanarayana [1]): A fuzzy ideal of

M is said to be a fuzzy prime ideal of M if is a not a constant function; and for

any two fuzzy ideals and of M, implies that either or .

5.2 Theorem (Th. 2.3 of Syam Prasad & Satyanarayana [1]): If is a fuzzy prime

ideal of M, then M = {x M / (x) = (o)} is a prime ideal of M.

5.3 Proposition (Syam Prasad [1]): Let I be an ideal of M and [0, 1).

Let be a fuzzy subset of M, defined by (x) = .

Then is a fuzzy prime ideal of M if I is a prime ideal of M.

5.4 Corollary (Syam Prasad [1]): Let I be an ideal of M. Then I (the

characteristic function of I) is a fuzzy prime ideal of M if and only if I is a prime

ideal of M.

5.5 Lemma (Lemma 2.6 of Syam Prasad & Satyanarayana [1]): If is a fuzzy

prime ideal of M, then (o) = 1.

5.6 Proposition (Th. 2.7 of Syam Prasad & Satyanarayana [1]): If is a fuzzy

prime ideal of M, then |Im | = 2.

6. Mislaneous concepts on -near-rings

Selvaraj & George [1] introduced the notion of strongly regular 2-primal -

near-rings and studied some characterizations of 2-primal and strongly 2-primal

-near-rings.

Selvaraj & George [2] gave some characterizations of left strongly regular

-near-rings. Also proved that in a weakly left duo -near-rings N, N is left

weakly -regular if and only if N is left strongly -regular.

Mustafa Uckun and Mehmet Ali Ozturk [1] studied the notion of

symmetric bi--Derivations, symmetric bi generalization -Derivations in -near-

rings.

6.1 Definition: Let M be a -near-ring and D(, ) a symmetric bi-additive

mapping of M. D(, ) is said to be a symmetric bi--derivation if

D(xy, z) = D(x, z)y + xD(y, z) for all x, y, z M and .

Then, for any y M, a mapping x ↦ D(x, y) is a -derivation.

Considering M as a 2-torsion free 3-prime left gamma-near-ring with

multiplicative centre C, Mustafa Uckun and Mehmet Ali Ozturk [1] studied

the trace of symmetric bi-gamma-derivations (also symmetric bi-generalized

gamma-derivations) on M.

6.2 Theorems (Mustafa Uckun and Mehmet Ali Ozturk [1]): Let D(.,.) be a

non-zero symmetric bi-gamma-derivation of M and F(.,.) a symmetric bi-additive

mapping of M. Let d and f be traces of D(.,.) and F(.,.), respectively.

In this case

(1) If d(M) is a subset of C, then M is a commutative ring.

(2) If d(y), d(y) + d(y) are elements of C(D(x,z)) for all x, y, z in M, then M is a

commutative ring.

(3) If F(.,.) is a non-zero symmetric bi-generalized gamma-derivation of M

associated with D(.,.) and f(M) is a subset of C, then M is a commutative ring.

(4) If F(.,.) is a non-zero symmetric bi-generalized gamma-derivation of M

associated with D(.,.) and f(y), f(y) + f(y) are elements of C(D(x,z)) for all x, y, z in

M, then M is a commutative ring.

Acknoledgements

I thank the authorities of Periyar Maniammai University for giving me an

opportunity to deliver this Lecture to the Faculty and Scholars of the

DEPARTMENT OF MATHEMATICS.

References

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