Keywords: Graded Gamma Rings and Modules, Gamma Anneids and Modu
Keywords: Graded Gamma Rings and Modules, Gamma Anneids and Modu
Keywords: Graded Gamma Rings and Modules, Gamma Anneids and Modu
Abstract
We introduce graded gamma rings from the most general point of view
via methods developed by Krasner and Halberstadt for Krasners general graded rings. We propose three equivalent aspects of studying
graded gamma rings. The graded Jacobson radical of a graded gamma
ring is introduced and its elementwise description is given. Also, a
relation between the graded Jacobson radical and the Jacobson radical
of a graded gamma ring is examined.
Keywords: Graded gamma rings and modules, Gamma anneids and moduloids, Jacobson radical
2010 Mathematics Subject Classification Primary 16Y99, Secondary 16W50,
16N20
Introduction
Gamma rings were introduced by Nobusawa [13] as an algebraic tool for observing the relationship between the groups of homomorphisms hom(B, C)
and hom(C, B) of commutative groups B and C. Let us recall the notion of
a -ring and a -ring in the sense of Nobusawa, following [1].
If R and are abelian additive groups, then R is called a -ring if, for all
x, y, z R and , , the following conditions are satisfied:
i) xy R;
ii) (x+y)z = xz +yz, x(+)z = xz +xz, x(y +z) = xy +xz;
iii) (xy)z = x(y)z.
It is called a -ring in the sense of Nobusawa if moreover we have:
i ) x ;
ii ) (xy)z = x(y)z = x(y)z;
iii ) if xy = 0 for all x, y R, then = 0.
Later on, many mathematicians gave their contributions to the theory of
gamma rings, and mainly concerning radical theory (see e.g. [2, 3, 15, 12, 16]
and references therein). In [4] a group graded gamma ring is defined to be a
-ring R which is the direct sum of additive subgroups Rg , g K, such that
Rg Rh Rgh , for all g, h K, where K is a group. However, the following
example motivates us to generalize the previous notion. Let us suppose we
L
L
have graded S-modules B and C, B = xK Bx , C = xK Cx , where S
is a K-graded ring, K a group and let
M
R = HOMS (B, C) =
HOMS (B, C)x ,
xK
where HOMS (B, C)x is a group of graded morphisms of degree x [14], that
is, of S-linear mappings : B C such that (Bg ) Cxg , for all g K.
Similarly, observe = HOMS (C, B). Clearly, R is a -ring. If 1 Rx ,
2 Rz and y , then 1 2 (Bg ) 1 ((Czg )) 1 (Byzg ) Cxyzg ,
and hence, 1 2 Rxyz . So, one way to expand the notion of a group
graded gamma ring is to require
Rg f Rh Rgf h ,
for all f, g, h K, if =
gK
(1.1)
(k K) Rg f Rh Rk ,
(1.2)
L
L
for all f, g, h K, the additive graduations R = gK Rg and = gK g
of R and , respectively, and the structure of a -ring R will imply operation
(generally partial) in K. Indeed, if Rg f Rh 6= {0}, then k K, for which
Rg f Rh Rk , is unique, and is arbitrary if Rg f Rh = {0}. So, if Rg f Rh 6=
{0}, we may define gf h := k. Of course, for those g, f, h K for which
Rg f Rh = {0}, this ternary operation may be defined arbitrarily in order
to make it defined everywhere on K. So, there is no need to assume anything
of K except of being a nonempty set. We do not stop here, but
L go further
by assuming
different
sets
of
grades
of
R
and
,
namely
R
=
gK Rg and
L
= hH h . In that case, condition
(g, g K)(h H)(k K) Rg h Rg Rk
(1.3)
0 0
0 0
R 0
0 V
0 0
. Indeed, if R1 =
, R2 =
, R3 =
,
0 S
0 0
0 0
W 0
0 0
R4 =
, then R(M ) = R1 R2 R3 R4 and = R1 R4 , and
0 S
for all i, j {1, 2, 3, 4} and k {1, 4} there exists l {1, 2, 3, 4} such that
Ri Rk Rj Rl .
2. Semidirect sums of rings.
If R is a ring which contains a subring S and ideal I such that R = S I,
then R is called the semidirect sum of S and I (see e.g. [7]).
Let R1 = S and R2 = I. Then, by putting = S, we get that R = R1 R2 is
a graded -ring since RSR R, R2 R2 = ISI I = R2 , R2 R1 = ISS
I = R2 , R1 R2 = SSI I = R2 , R1 R1 = SSS S = R1 .
In particular, the Dorroh extension may be regarded as a graded gamma
ring in our sense.
Theory we are about to establish covers the theory of ordinary -rings and
of group graded -rings. Also, as -rings generalize the concept of a ring,
graded -rings will provide a generalization of a graded ring. Before observing graded -rings, we deal with preliminaries regarding Krasners definition
of a graduation. If R is a graded -ring, A the homogeneous part of R and
G the homogeneous part of , we prove that a -ring R is determined by
A and G. We also define and give an elementwise description of the graded
Jacobson radical of a graded -ring R in terms of G and A. Finally, the
comparison of the graded Jacobson and the Jacobson radical of a graded
-ring is given.
On Krasners graduations
(2.1)
L
such that G =
G , where Sg(G) is the set of all subgroups of G, is
called a graduation. A group with a graduation is called a graded group. A
graduation is called strict if G 6= {1}, for all . If = { | G 6=
{1}}, then = | is a strict graduation of G and is called a strict kernel
of graduation (2.1). Elements are called grades S
and the corresponding
S
G are called homogeneous components. The set H = G (= G
if G 6= {1}) is called the homogeneous part of a graded group G and elements
x H are called homogeneous. The unique for which 1 6= x H
belongs to G is called the grade of x and is denoted by (x).
Element 1 generally speaking does not have a grade and \ are
called empty grades. However, we may associate a grade from \ to 1,
which we denote by 0 and call it a zero grade. If = {0}, and if we
put (1) = 0, the graduation is called proper. Throughout this paper we
assume all graduations to be proper.
Let G be a graded group with graduation (2.1) and let
: Sg(G), ( ) = G ( )
(2.2)
ii) x H x1 H;
iii) x, y, z, xy, yz H y 6= 1 xz H;
iv) x, y H xy
/ H xy = yx;
v) H generates G;
vi) If n 2 and if the elements x1 , . . . , xn H are such that for all
i, j {1, . . . , n}, i 6= j, xi xj
/ H, then x1 . . . xn 6= 1.
Multiplicative operation on G induces a partial operation in H. Namely, if
x, y H, then xy is defined in H if and only if xy G is the element from
H, and in that case the result is the same and we write it the same way. If
this situation occurs, we say that elements x, y are composable (addible in
case of an additive operation) and we write x#y. Clearly, x#y if and only
if x, y come from the same subgroup G(a) = {x H | ax H}, a H .
In case when H with the induced operation from G is given, we may reconstruct G up to H-isomorphism. Indeed, if a H , then G(a) may be
defined as G(a) = {x H | a#x}. G is then the direct sum of different
subgroups G(a) and H is obviously the homogeneous part of G. The group
G which is obtained in this way is called the
L linearization of the structure
(H, ) and we denote it by H. Hence, H = aH G(a).
There is a natural idea now to define the graded group from the so called
homogeneous aspect using the corresponding homogeneous part, at least up
to isomorphism. We need to characterize the structure (H, ), which is the
homogeneous part of some graded group, with operation induced from that
group. This characterization is the subject of the following theorem.
Theorem 2.3. The structure (H, ) is the homogeneous part of some graded
group G, with operation induced from that group, if and only if the following
conditions hold:
i) (1 H)(x H) x#1 x1 = x;
ii) (x H) x#x;
iii) (x, y, z H) x#y y#z y 6= 1 x#z;
iv) For all a H for which a 6= 1, H(a) = {x H | a#x} is a group.
Definition 2.4. The structure (H, ) which satisfies conditions of Theorem
2.3 is called a homogroupoid.
Let H be a homogroupoid and the set of groups H(a) = {x H | ax
H}, a H = H \ {1}, and let us denote elements of by , , , . . . . If
, we denote by H() or by G or by H the corresponding group
H(a) which defines . Hence, if a H , then we define its grade as (a) :=
A subhomogroupoid K of a homogroupoid H is normal if each addibility group K(a) of K is normal subgroup of H(a), where a H . If K is
a normal subhomogroupoid of H, then the homogeneous part of H/K is
S
aH H(a)/H(a) K.
Definition 2.7. A homomorphism f : G G of graded groups with
homogeneous parts H and H , respectively, is called quasihomogeneous if
f (H) H . A quasihomogeneous homomorphism f is called a homogeneous
homomorphism if for x, y H with f (x) 6= 1 , f (y) 6= 1 , we have
(f (x)) = (f (y)) (x) = (y).
The homogeneous counterpart of the above definition follows.
Definition 2.8. Let H and H be homogroupoids. The mapping f : H
H is called quasihomomorphism if x#y implies f (x)#f (y) and f (xy) =
f (x)f (y), x, y H. A quasihomomorphism is called homomorphism if the
composability of nonidentity images implies the composability of the originals.
As we have seen, homogroupoids are homogeneous parts of graded groups
with induced partial operation. Analogously, an anneid and a moduloid
represent homogeneous parts of a Krasner graded ring and a graded module, respectively, with induced operations. A ring R is called graded if
(R, +) is a commutative graded group in the above sense with graduation
R : Sg(R, +), R () = R , and if for all , , R R R , for
some , while a right R-module M is graded if R is a graded ring
and M a commutative graded group with graduation M : D Sg(M, +),
M (d) = Md , and if for all and d D there exists t D such that
Md R Mt . As an example of a Krasner graded ring one can take for
instance every group graded ring in the usual sense. However, there are
important types of rings which are graded in Krasners sense, but not necessarily group graded, such as the already mentioned ideal extension.
If M and M are A-moduloids, than the mapping f : M M is called
Graded -rings
3.1
Nonhomogeneous aspect
Let R be a -ring in the sense of Nobusawa and let R and be graded groups
in theLsense of Krasner with graduations R : Sg(R,L
+), R () = R ,
R = R , and : D Sg(, +), (d) = d , = dD d , respectively. Also, let A be the homogeneous part of R and G the homogeneous
part of . If the following conditions are satisfied:
(, )(d D)( ) R d R R ,
(3.1)
(3.2)
3.2
Semihomogeneous aspect
3.3
Homogeneous aspect
10
L
L
L
L
are hence A = aA A(a) and dD Gd = G G(),
Lrespec {0} and D = D {0}, A =
tively. Naturally,
if
we
put
A
L
and G = dD Gd .
Remark 3.5. Gamma anneid could also be called a gamma ringoid but we
kept the term which resembles the original notion of the homogeneous part
of a graded ring in Krasners sense, the anneid.
The following theorem gives us the characterization of gamma anneids.
Theorem 3.6. Let A and G be commutative homogroupoids. Then A is a
G-anneid of Nobusawa if and only if:
i) (x, y A)( G) xy A;
ii) (, G)(x A) x G;
iii) (a, a , b, b A)(, G) a#a ab#a b (a + a )b = ab +
a b, # ab#a b a( + )b = ab + a b, b#b ab#ab
a(b + b ) = ab + ab ;
iv) (a, b, c A)(, G) (ab)c = a(bc) = a(b)c ab = 0
= 0.
Proof. It is clear that the givenLconditions are necessary. Now,
L suppose that
i) iv) hold. Let R = A =
G
=
A(a)
and
aA
G G()
Pand
let
x
,
y
R,
i
=
1,
2,
.
Then
x
,
y
and
are
of
the
form
xi ,
i
P i
P
y and
d , respectively, where xi , y A = {x A | x = 0 (x 6=
0 (x) = )}, d Gd = { G | = 0 ( 6= 0 d() = d)},
d =
xi = y = 0 and
P P
P0 for all but finitely many and d, respectively. Now
put x
i
y := d xi d y . Then
X
X X
(
x1 + x
2 )
y =
(x1 + x2 )
d
y
XXX
x1 d y +
XXX
x2 d y
= x
1
y + x
2
y.
We analogously prove the other properties which make R a graded -ring
of Nobusawa.
Hence, given a G-anneid A, R = A, and = G, R is a graded -ring of type
(, D), where, = {0} is the set of grades of R, and D = D {0}
the set of grades of . Let us recall once again, elements of are denoted
by , , , . . . , but is actually comprised of addibility groups A(a) = {x
A | a + x A}, a 6= 0. The grade (a) of 0 6= a A is defined to be
the element which denotes A(a). Conversely, given , the
corresponding addibility group will be denoted by A(a) if (a) = , or it will
11
Graded -modules
(4.1)
12
13
P
For future purposes, let us mention that, generally, when we write nj=1 aj ,
where aj are elements of a G-anneid A, it is assumed that aj are mutually
addible.
If I and J are right (left, two-sided) ideals of a G-anneid A, then we define
I + J to be the set of all x A such that x = a + b, where a I, b J. It
is easy to verify that I + J is also a right (left, two-sided) ideal of A. It is
also an easy exercise to prove that the intersection of right (left, two-sided)
ideals of A is again a right (left, two-sided) ideal of A.
The smallest right ideal which contains an element a of a G-anneid A is
called the principal right ideal generated by a and is denoted by |ai. The
principal left ha| and the ideal hai generated by a are defined similarly.
Like in the case of gamma rings, it is clear that |ai = Za + aGA, ha| =
Za + AGa, and hai = Za + aGA + AGa + AGaGA.
If I is an ideal of a G-anneid A, then A/I is a G-anneid if we put
(a + I)(b + I) = ab + I,
(a + I, b + I A/I)
Jacobson radicals of -rings are extensively treated (see e.g. [3] and [15]),
and here, we will introduce and observe these radicals for gamma anneids by
following the concepts and results obtained for rings in [6] and for anneids in
[5] and in the final section we will obtain the relations among those radicals.
All results can be easily translated to graded gamma rings.
Let M be an AG-moduloid, N an AG-submoduloid and let S be a subset of
M. If we define the set (N : S) to be the set of a A such that SGa N,
then (N : S) is a right ideal of A. Indeed, let b, b (N : S), b#b , , G,
a A and s S. Then SG(bb ) = SGbSGb N and (sb)a N a
N.
Lemma 6.1. If M is a regular AG-moduloid and x M, G, then
A/(0 : x)
= xA, where (0 : x) = {a A | xa = 0}.
Proof. Let : A xA be the mapping defined with (a) = xa, : G
G and : A A identities. Obviously, (, , ) is a homomorphism and
ker = (0 : x) .
14
G-anneid A is called primitive if there exists an irrefaithful AG-moduloid M ((0 : M ) = {0}). It is called
there exists an irreducible and faithful AG-moduloid
not).
15
Proof. Let M be a strictly cyclic regular AG-moduloid with a strict generator x. Since M = xA
= A/(0 : x) , we only need to prove that (0 : x)
is modular. Since x xA and since M is regular, there exists a A such
that x = xa. Let b A. Then xb = (xa)b. If xb = x(ab) = 0, then
both b and ab belong to (0 : x) . If xb = x(ab) 6= 0, then, since M
is regular, b#ab and hence x(b ab) = 0, so b ab (0 : x) . Thus,
(0 : x) is modular.
Conversely, if I is a right modular ideal of a G-anneid A with a as an
-left identity modulo I, then a + I is an -strict generator of A/I and
I = (0 : a + I) .
Corollary 6.12. If I is a right modular ideal of A, then I contains (I : A).
Following propositions can be proved by similar arguments given for the
case of rings [6].
Proposition 6.13. Every right modular ideal is contained in a maximal
right ideal.
Proposition 6.14. An AG-moduloid M is irreducible if and only if:
a) M 6= {0};
b) Every nonzero element of M is a strict generator of M.
Proposition 6.15. A regular AG-moduloid M is irreducible if and only if
M
= A/I for some right modular maximal ideal I of A.
Corollary 6.16. Every primitive ideal J of a G-anneid A has the form
(I : A) for some right modular maximal ideal I of A. Conversely, if I is
a right modular maximal ideal of a G-anneid A, then (I : A) is a large
primitive ideal (primitive if A is right regular).
Theorem 6.17. The Jacobson radical of a right regular G-anneid is the
intersection of its right modular maximal ideals.
16
Definition 7.2. The grade of all left identities modulo I is called the grade
of I.
Let e be an element whose grade (e) is an -idempotent of , that
is, (e)d()(e) = (e). Notice that A(e) is then an {}-ring. Indeed, let
x, y A(e). Then x#e, y#e and e#ee, hence, xy#e (see Lemma 3.1).
For the sake of simplicity, we will denote (e) also by e in the sequel.
Theorem 7.3. Let A be a regular G-anneid and e an -idempotent of .
To every right modular maximal ideal I of A with grade e, let us assign
Ie = I A(e). Also, to every right maximal ideal S of A(e), let us assign the
set S of elements x A such that xGA A(e) S. This establishes a one
to one correspondence between the set of right modular maximal ideals of A
with grade e and the set of right modular maximal ideals of A(e).
Proof. We will follow the arguments used for anneids in [5]. Let S be a right
modular maximal ideal of A(e) and let u be an -left identity modulo S. It
is clear that S is a right ideal of A. Let s S, a A, G, and assume
that 0 6= sa A(e). Then
ed()e = e = (sa) = (s)d()(a).
Since (s) = e, and since A is regular, it follows that (a) = e. Hence,
sa S, which means that S S A(e). Conversely, let x S A(e)
be such that x
/ S. Since S is maximal, we have A(e) = S + |xi, and
so u = s + nx + xa, where s S, n Z, G, a A. Since xa,
xx xGA A(e) S and ux = sx + nxx + xax, we have ux S,
and since u is an -identity modulo S, x S. Thus, S A(e) S. We have
proved that S A(e) = S.
Let a A be
Next we want to prove that u is an -left identity modulo S.
17
18
19
Acknowledgement
The author would like to express his deepest gratitude to Academician Professor Mirjana Vukovic who introduced him to the theory of general graded
rings.
References
[1] W. E. Barnes, On the -rings of Nobusawa, Pacific J. Math. 18 (1966),
411422.
[2] G. L. Booth, N. J. Groenewald, -rings and normal radicals, Period.
Math. Hung. 31 (1) (1995), 510.
[3] W. E. Coppage, J. Luh, Radicals of gamma rings, J. Math. Soc. Japan
21 (1971), 4052.
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