2. Graded 1-Absorbing Prime Ideals
In this section, we introduce and examine the concept of graded 1-absorbing prime ideals over non-commutative graded rings. From [
7], we have the fact that a proper graded ideal
P of a commutative graded ring
R is called a graded 1-absorbing prime ideal if whenever
for some nonunits
, either
or
. In what follows,
R is a non-commutative graded ring with nonzero unity unless indicated otherwise. We have the following:
Definition 1. Let R be a graded ring. A proper graded ideal P of R is called graded 1-absorbing prime if for all nonunit elements such that , we have either or .
For commutative graded rings, the two concepts of graded 1-absorbing prime ideals coincide. We now give an example to show that for non-commutative graded rings, it is not the same case:
Example 1. Consider (the ring of all matrices with integer entries) and . Then, R is G-graded by , , and . Since is a graded prime ideal of R, P is a graded 1-absorbing prime ideal of R. On the other hand, , , and are nonunit elements such that , , and .
Lemma 1. Let P be a graded 1-absorbing prime ideal of a graded ring R. If J is a proper graded ideal of R and are nonunit elements such that , then either or .
Proof. Suppose that and . Then, for all as J is a graded ideal. Let . Then, is a nonunit element as J is proper. Now, , which implies that as P is a graded 1-absorbing prime and . So, for all , and hence . Thus, . □
Theorem 1. Suppose that P is a proper graded ideal of a graded ring R. Then, P is a graded 1-absorbing prime ideal of R if and only if whenever , for some proper graded ideals of R, either or .
Proof. Suppose that P is a graded 1-absorbing prime ideal of R and for some proper graded ideals of R such that . Then, there are nonunit elements and such that , and then there are such that . Note that as are graded ideals, and . Since and , it follows from Lemma 1 that . Conversely, suppose that for some nonunit elements and . Assume that , and . Then are proper graded ideals of R with and . Hence, , and thus . □
Let
R and
S be two
G-graded rings. Then, a ring homomorphism
is said to be a graded ring homomorphism if
for all
[
1].
Theorem 2. Let be a graded ring epimorphism such that is a nonunit in S for every nonunit element r in R.
- 1.
If K is a graded 1-absorbing prime ideal of S, then is a graded 1-absorbing prime ideal of R.
- 2.
If P is a graded 1-absorbing prime ideal of R with , then is a graded 1-absorbing prime ideal of S.
Proof. - 1.
Suppose that for some nonunit elements . Then, , which means that or . It follows that or . Hence, is a graded 1-absorbing prime ideal of R.
- 2.
Suppose that for some nonunit elements . Since f is surjective, there exist nonunit elements such that , , and . Therefore, . Since , we conclude that . Thus, or , and so or . Hence, is a graded 1-absorbing prime ideal of S.
□
Let
R be a
G-graded ring and
P be a graded ideal of
R. Then,
is a
G-graded ring by
, for all
[
1].
Corollary 1. Suppose that P and K are proper graded ideals of a graded ring R with and . Then, K is a graded 1-absorbing prime ideal of R if and only if is a graded 1-absorbing prime ideal of .
Proof. Define by . Then, f is a graded ring epimorphism, , is a nonunit for every nonunit , and . Suppose that K is a graded 1-absorbing prime ideal of R. Then, is a graded 1-absorbing prime ideal of by Theorem 2 (2). Conversely, is a graded 1-absorbing prime ideal of R by Theorem 2 (1). □
Definition 2. Let R be a graded ring, , and P be a graded ideal of R such that . Then,
- 1.
P is called a g-1-absorbing prime if for all nonunit elements such that , either or .
- 2.
P is called a g-prime if for all nonunit elements such that , either or .
Theorem 3. Let R be a graded ring. Suppose that R has a g-1-absorbing prime ideal that is not g-prime. If such that α is a nonunit element and β is a unit element, then is a unit element.
Proof. Let P be a g-1-absorbing prime ideal of R that is not g-prime. Then, there are nonunit elements such that and . Since , it follows that . Also, . If is a nonunit element, then . Hence, , and then since is a unit element, we have , which is a contradiction. Hence, is a unit element. □
Theorem 4. Let R be a graded ring such that is not a local ring. Then, every e-1-absorbing prime ideal of R is e-prime.
Proof. Suppose that
R has an
e-1-absorbing prime ideal that is not
e-prime. Then, by Theorem 3, for
such that
is a nonunit element and
as a unit element, we have
as a unit element, and then by ([
11], Lemma 4.1),
is a local ring, which is a contradiction. □
A proper graded ideal
M of a graded ring
R is said to be a graded maximal ideal if whenever
P is a graded ideal of
R such that
, either
or
. A graded ring
R is said to a graded local ring if it has a unique graded maximal ideal [
1].
Lemma 2. Let R be a graded local ring with unique graded maximal ideal M. Then, P is a graded 1-absorbing prime ideal of R if and only if P is a graded prime ideal of R or .
Proof. Suppose that P is a graded 1-absorbing prime ideal of R that is not graded prime. Clearly, . Since P is not graded prime, there are such that . Let . Then, for all as M is a graded ideal. Also, , which implies that for all . Again, we have for all . Thus, , and hence . Conversely, if P is a graded prime ideal, then P is a graded 1-absorbing prime ideal. Suppose that . Then, clearly, P is proper. Let such that . Then, . Therefore, P is a graded 1-absorbing prime ideal of R. □
Proposition 1. Let R be a graded local ring with unique graded maximal ideal M. Then, R has a graded 1-absorbing prime ideal that is not graded prime if and only if .
Proof. Suppose that R has a graded 1-absorbing prime ideal P of R that is not graded prime. Then, by Lemma 2, , and hence . Conversely, it is clear that , so is proper and there exists , and then as M is a graded ideal, there exists such that . But , so is not a graded prime ideal of R. Let such that . Then, , and hence is a graded 1-absorbing prime ideal of R. □
Proposition 2. Let R be a graded local ring. If P and K are two graded 1-absorbing prime ideals of R that are not graded prime, then and are graded 1-absorbing prime ideals of R.
Proof. Let M be a unique graded maximal ideal of R. Then, by Lemma 2, . Let such that . Then, , and hence is a graded 1-absorbing prime ideal of R. Similarly, is a graded 1-absorbing prime ideal of R. □
A graded ring
R is said to be graded prime if
is a graded prime ideal of
R [
4].
Theorem 5. Let R be a graded local ring with unique graded maximal ideal M. Then, is a graded 1-absorbing prime ideal of R if and only if R is graded prime or .
Proof. The result holds directly from Lemma 2. □
Assume that
R is a
G-graded ring and
M is an
R-module. Then,
M is said to be
G-graded if
with
for all
, where
is an additive subgroup of
M for all
. The elements of
are called homogeneous of degree
g. It is clear that
is an
-submodule of
M for all
. We assume that
. Let
N be an
R-submodule of a graded
R-module
M. Then,
N is said to be a graded
R-submodule if
; i.e., for
,
, where
for all
. It is known that an
R-submodule of a graded
R-module is not necessarily graded. Let
M be an
R-module. The idealization
of
M is a ring with componentwise addition and multiplication;
and
for each
and
. By ([
11], Remark 3.1),
is a nonunit in
if and only if
r is a nonunit in
R. Let
G be an abelian group and
M be a
G-graded
R-module. Then,
is
G-graded by
for all
[
12]. Moreover, if
P is an ideal of
R and
N is an
R submodule of
M such that
, then
is a graded ideal of
if and only if
P is a graded ideal of
R and
N is a graded
R-submodule of
M ([
12], Proposition 3.3).
Theorem 6. Let M be a graded R-module and P be a proper graded ideal of R. Then, is a graded 1-absorbing prime ideal of if and only if P is a graded 1-absorbing prime ideal of R.
Proof. Suppose that is a graded 1-absorbing prime ideal of and let for some nonunits . Then, are nonunits in with , and then or . Hence, or . Thus, P is a graded 1-absorbing prime ideal of R. Conversely, let for some nonunits and . Then, , and then or . If , then , and if , then . Hence, is a graded 1-absorbing prime ideal of . □
3. Graded Weakly 1-Absorbing Prime Ideals
In this section, we introduce and examine the concept of graded weakly 1-absorbing prime ideals over non-commutative graded rings.
Definition 3. Let R be a graded ring. A proper graded ideal P of R is called graded weakly 1-absorbing prime if for all nonunit elements such that , we have either or .
Clearly, every graded 1-absorbing prime ideal is graded weakly 1-absorbing prime. However, the next example shows that a graded weakly 1-absorbing prime ideal is not necessarily graded 1-absorbing prime:
Example 2. Consider and . Then, R is G-graded by , , and . Clearly, is a graded weakly 1-absorbing prime ideal of R. On the other hand, P is not graded 1-absorbing prime, since , , and are nonunit elements such that , , and .
Proposition 3. Let R be a graded ring and K be a graded weakly 1-absorbing prime ideal of R. If P is a graded ideal of R with , then is a graded weakly 1-absorbing prime ideal of .
Proof. Suppose that for some nonunits . Then, are nonunits such that , and then either or , which implies that either or . Hence, is a graded weakly 1-absorbing prime ideal of . □
Proposition 4. Let be proper graded ideals of a graded ring R such that . If P is a graded weakly 1-absorbing prime ideal of R and is a graded weakly 1-absorbing prime ideal of , then K is a graded weakly 1-absorbing prime ideal of R.
Proof. Let be nonunits such that . If , then either or . Suppose that . Then, are nonunits such that , and then either or , which implies that either or . Hence, K is a graded weakly 1-absorbing prime ideal of R. □
Let
be a graded ring homomorphism. Then, by ([
10], Proposition 5),
is a graded ideal of
S.
Theorem 7. Let be a graded ring epimorphism.
- 1.
If P is a graded weakly 1-absorbing prime ideal of R and , then is a graded weakly 1-absorbing prime ideal of S.
- 2.
If Q is a graded weakly 1-absorbing prime ideal of S, is a nonunit in S for every nonunit r in R, and is a graded weakly 1-absorbing prime ideal of R, then is a graded weakly 1-absorbing prime ideal of R.
Proof. - 1.
Since P is a graded weakly 1-absorbing prime ideal of R and , we conclude that is a graded weakly 1-absorbing prime ideal of by Proposition 3, and then the result follows since is isomorphic to S.
- 2.
Let . Then, . Since is isomorphic to S, we conclude that is a graded weakly 1-absorbing prime ideal of . Since is a graded weakly 1-absorbing prime ideal of R and is a graded weakly 1-absorbing prime ideal of , we conclude that is a graded weakly 1-absorbing prime ideal of R by Proposition 4.
□
Definition 4. Let R be a graded ring, , and P be a graded ideal of R such that . Then, P is called g-weakly 1-absorbing prime if for all nonunit elements such that , either or .
Definition 5. Suppose that P is a graded weakly 1-absorbing prime ideal of a graded ring R and are nonunit elements.
- 1.
We say that is a homogeneous triple-zero (htz for short) of P if , , and .
- 2.
Suppose that for some graded ideals of R. Then, we say that P is a free htz with respect to if is not an htz of P, for every , and .
Definition 6. Suppose that P is a g-weakly 1-absorbing prime ideal of a graded ring R and are nonunit elements.
- 1.
We say that is a g-triple-zero (g-tz for short) of P if , , and .
- 2.
Suppose that for some graded ideals of R. Then, we say that P is a free g-tz with respect to if is not a g-tz of P, for every , and .
Clearly, if P is a graded weakly 1-absorbing prime ideal that is not graded 1-absorbing prime, then there exists an htz of P. Also, if P is a g-weakly 1-absorbing prime ideal that is not g-1-absorbing prime, then there exists a g-tz of P. So, we introduce the following example that is motivated by Example 2:
Example 3. Consider and . Then, R is G-graded by , , and . Then, by Example 2, is a graded weakly 1-absorbing prime ideal of R that is not graded 1-absorbing prime, and P is a 0-weakly 1-absorbing prime ideal of R that is not 0-1-absorbing prime. Note that , , are nonunit elements such that , and . Hence, are htzs of P, and is a 0-tz of P.
Lemma 3. Let P be a g-weakly 1-absorbing prime ideal of a graded ring R and suppose that is a g-tz of P for some nonunit elements . Assume that the sum of every two nonunit elements of is nonunit. Then,
- 1.
.
- 2.
.
- 3.
.
- 4.
.
- 5.
.
- 6.
.
Proof. - 1.
Suppose that . Then, there exist and such that . Now, . Hence, . Since , we have and consequently , which is a contradiction.
- 2.
Suppose that . Then, there exist and such that . Now, . Hence, . Since , we have and then , which is a contradiction.
- 3.
Suppose that . Then, there exists such that . Now, . Hence, . Since , we have and then , which is a contradiction.
- 4.
Suppose that . Then, there exist such that . Now, by (2) and (3). Hence, . Since , we have and then , which is a contradiction.
- 5.
Suppose that . Then, there exist such that . Now, by (1) and (3). Hence, . So we have or . Hence, or , which is a contradiction.
- 6.
Suppose that . Then, there exist such that . Now, by (1) and (2). Hence, . So we have or . Hence, or , which is a contradiction.
□
Theorem 8. Let R be a graded ring and such that the sum of every two nonunit elements of is a nonunit. Suppose that P is a graded ideal of R such that . Then, P is a g-weakly 1-absorbing prime ideal of R if and only if P is a g-1-absorbing prime ideal of R.
Proof. Suppose that P is a g-weakly 1-absorbing prime ideal of R that is not a g-1-absorbing prime. Then, P has a g-tz for some nonunits . Assume that for some . Then, by Lemma 3, we have . Hence, . So, we have either or , and thus either or , which is a contradiction. Hence, P is a g-1-absorbing prime ideal of R. The converse is clear. □
Corollary 2. Let R be a graded ring and such that the sum of every two nonunit elements of is a nonunit. If P is a g-weakly 1-absorbing prime ideal of R that is not g-1-absorbing prime, then .
Theorem 9. Let R be a graded ring and such that the sum of every two nonunit elements of is a nonunit. Suppose that P is a g-weakly 1-absorbing prime ideal of R such that . Then, P is a g-1-absorbing prime ideal of R.
Proof. Now, , and then it follows from Theorem 8 that P is a g-1-absorbing prime ideal of R. □
Proposition 5. Let R be a graded ring and such that the sum of every two nonunit elements of is a nonunit. Suppose that such that . Then, the graded left ideal is g-1-absorbing prime if and only if is a g-weakly 1-absorbing prime left ideal of R.
Proof. Suppose that is a g-weakly 1-absorbing prime left ideal of R that is not g-1-absorbing prime. Then, has a g-tz for some nonunit elements . Now, . If , then , and then , which is a contradiction. Hence, , and so , which implies that , and then , which is a contradiction. Therefore, is g-1-absorbing prime. The converse is clear. □
A
G-graded ring
R is said to be a cross-product if
contains a unit for all
[
1]. Let
S and
T be two
G-graded rings. Then,
is a
G-graded ring by
for all
[
1].
Theorem 10. Let S and T be two graded rings such that T is a cross-product. Let and P be a proper graded ideal of S. If is a graded weakly 1-absorbing prime ideal of R, then P is a graded 1-absorbing prime ideal of S.
Proof. Suppose that for some nonunit elements . Then, , and for some . Choose unit elements , and . Then, , and so either or . Hence, either or . Thus, P is a graded 1-absorbing prime ideal of S. □
A graded ring
R is said to be a graded division ring if every nonzero homogeneous element of
R is a unit [
1].
Theorem 11. Let S and T be two graded rings that are cross-products but not graded division rings. Let and P be a nonzero proper graded ideal of R. If P is a graded weakly 1-absorbing prime ideal of R, then for some graded prime ideal of S or for some graded prime ideal of T.
Proof. Now, for some graded ideals of S and of T. Since P is nonzero, or . Without loss of generality, we may assume that . Then, there exists , and then , for some . Note that as is a graded ideal. Choose a unit element , where ; we conclude or . Then, we have or . Assume that . Now, we will show that is a graded prime ideal of T. Let for some . If y or z is a unit element, then we have or . So, assume that are nonunit elements in T. Since S is not a graded division ring, there exists a nonzero nonunit . Note that , and for some , so choose unit elements , and . This implies that . We conclude that or . Thus, we get or , and so is a graded prime ideal of T. In another case, one can similarly show that and is a graded prime ideal of S. □
Theorem 12. Let S, T, and L be graded rings such that they are cross-products. Let and . If is a nonzero graded weakly 1-absorbing prime ideal of R, then is a graded 1-absorbing prime ideal of H or is a graded 1-absorbing prime ideal of L.
Proof. Since P is nonzero, there exists , and then or or for some . Choose unit elements , and . So, , and then either or . Hence, either and , or , and so or . Thus, by Theorem 10, is a graded 1-absorbing prime ideal of H or is a graded 1-absorbing prime ideal of L. □
Proposition 6. Let R be a graded local ring with unique graded maximal ideal M. If , then every proper graded ideal of R is a graded weakly 1-absorbing prime ideal of R.
Proof. Clearly, is a graded weakly 1-absorbing prime ideal of R. Let P be a nonzero proper graded ideal of R. Assume that P is not a graded weakly 1-absorbing prime ideal. Then, there exist nonunit elements such that with and . Since are nonunits, they are elements of M. So, , which is a contradiction. □