A Note On Rhotrix
A Note On Rhotrix
A Note On Rhotrix
New Theory
ISSN: 2149-1402
Article History Abstract − In this paper, we define algebraic operations on 3-dimensional rhotrices
Received : 29.03.2019 over an arbitrary ring R and show that the set of 3-dimensional rhotrices over an
arbitrary ring R is a ring according to these operations. We investigate the properties
Accepted : 02.11.2019
of a rhotrices ring. Furthermore, we characterize the ideals of a rhotrices ring. Also,
Published : 30.12.2019 maximal ideals and prime ideals of a rhotrices ring are investigated. An example of
Original Article these concepts is presented.
1. Introduction
The concept of the rhotrix is a mathematical structure in the rhombodial form of real numbers defined
by Atanasov and Shannon [1], inspired by the concepts of matrix tertion and matrix netrion. In 2003,
Ajibade [2] defined an object that lies between 2 × 2 dimensional matrices and 3 × 3 dimensional
matrices called rhotrix as follows:
Definition 1.1. [2] Let a, b, c, d, e be real numbers. Then a mathematical rhombodial form
* a +
R= b c d
e
is called 3 − dimensional rhotrix over real numbers. The entry c in rhotrix R is called the heart of
R denoted by h(R).
In [3] it has been shown that the set of all three-dimensional real rhotrices together with the
* 0 +
operations addition (+) and multiplication () is a commutative ring with identity I = 0 1 0 .
0
* a +
Definition 1.3. Let R = b h(R) d be in R. If RQ = I such that there exists Q ∈ R then
e
Q is called the inverse of R, denoted by R−1 , and
* a +
−1 −1
Q=R = b −h(R) d where h(R) 6= 0.
h(R)2
e
Other multiplication of rhotrices called row-column multiplication was proposed by Sani [4]. This
multiplication is as follows:
* a + * f +
Definition 1.4. Let R = b h(R) d and Q = g h(Q) j be in R.
e k
* af + dg +
R•Q= bf + eg h(R)h(Q) aj + dk
bf + ek
Studies on this subject has progressed quickly after the rhotrix definition. Several authors have
obtained interesting results on 3-dimensional rhotrices. See [5] for a comprehensive survey of the
literature on these developments.
Journal of New Theory 29 (2019) 32-41 / Rhotrices Ring 34
The definition of n-dimensional rhotrix over an arbitrary ring was firstly given by Mohammed in [6]
and he gave the set of all rhotrices over an arbitrary ring is a ring together with the operations of
rhotrix addition and row-cloum rhotrix multiplication.
In this section, it has been shown that the set of 3-dimensional rhotrices over an arbitrary ring is
a ring with the operations rhotrix addition and “hearty multiplication” as different from multipli-
cation in Mohammed’s work [6]. Also we investigate the basic properties of the rhotrices ring.
Definition 2.1. Let (R, +, .) be a ring with identity. By a 3 − dimensional rhotrix over the ring R,
we mean a rhomboidal array defined by
* a +
A= b c d
e
where a, b, c, d, e are in the ring R. The entry c of A is called heart of A denoted by h(A).
The set of all 3-dimensional rhotrices over the ring R denoted by R3 (R),
(* a + )
R3 (R) = b c d a, b, c, d, e ∈ R
e
.
b and multiplication( ⊙) on R3 (R) by
We define two binary operations addition ( +)
* a + * a
′ + * a+a
′ +
b b′
b c d + c
′
d
′
= b+b
′
c+c
′
d+d
′
(1)
′ ′
e e e+e
* a + * a
′ + * ′
a.c + c.a
′ +
′ ′ ′ ′ ′ ′ ′ ′
b c d ⊙ b c d = b.c + c.b c.c d.c + c.d (2)
′ ′ ′
e e e.c + c.e
* a +* a
′ +
′ ′ ′
for all b c d , b c d ∈ R3 (R). It is easy to check that these operations are well defined,
′
e e
since ” + ” and ”.” in R are well-defined.
Theorem 2.2. The set of all 3 − dimensional rhotrices R3 (R) over the ring R is a ring with respect
b and ” ⊙ ”.
to operations ”+”
b is a commutative group. Now let’s show that the triple
Proof. It’s easy to see that (R3 (R), +)
b
(R3 (R), +, ⊙) is a ring.
* a + * a′ + * x +
For all P = b c d ,Q = b′ c′ d′ ,S= y z t ∈ R3 (R)
e e′ u
* ′
a.c + c.a
′ + * x +
′ ′ ′ ′ ′
(P ⊙ Q) ⊙ S = b.c + c.b c.c d.c + c.d ⊙ y z t
′ ′
e.c + c.e u
* ′ ′ ′
a.c .z + c.a .z + c.c .x +
′ ′ ′ ′ ′ ′ ′
= b.c .z + c.b .z + c.c .y c.c .z d.c .z + c.d .z + c.c .t
′ ′ ′
e.c .z + c.e .z + c.c .u
= P ⊙ (Q ⊙ S)
* a+a
′ + * x +
′ ′ ′
b
(P +Q) ⊙S = b+b c+c d+d ⊙ y z t
′
e+e u
* ′ ′
(a + a ).z + (c + c ).x +
′ ′ ′ ′ ′
= (b + b ).z + (c + c ).y (c + c ).z (d + d ).z + (c + c ).t
′ ′
(e + e ).z + (c + c ).u
* ′ ′
a.z + a .z + c.x + c .x +
′ ′ ′ ′ ′
= b.z + b .z + c.y + c .y c.z + c .z d.z + d .z + c.t + c .t
′ ′
e.z + e .z + c.u + c .u
* a.z + c.x + * ′ ′
a .z + c .x +
′ ′ ′ ′ ′
= b.z + c.y c.z d.z + c.t + b b .z + c .y c .z d .z + c .t
′ ′
e.z + c.u e .z + c .u
b
= (P ⊙ S)+(Q ⊙ S)
b
and similarly it is easy to check that P ⊙ (S +Q) = (P ⊙ S)+(Pb ⊙ Q)
Thus < R3 (R), +,b ⊙ > is a ring.
Furthermore if R is a commutative ring, then R3 (R) is a commutative ring and if R is a ring with
* 0R +
identity 1R , then R3 (R) to be a ring with identity 1R3 (R) = 0R 1R 0R .
0R
Example 2.3. Let R = Z2 . R3 (R) is a rhotrix ring and since Z2 is a commutative ring, R3 (Z2 ) is a
commutative ring.
The following theorem give us the characteristic of the ring R3 (R) depends on the characteristic
of the ring R.
Theorem 2.4. The characteristic of the ring R3 (R) is equal to characteristic of the ring R.
Proof. Let R be a ring with CharR = k. Then the characteristic of the ring R3 (R) is k. Let
CharR3 (R) = t, we show that k = t.
a *
+ * a +
CharR3 (R) = t ⇒ t. b c d = 0R3 (R) , f or all b c d ∈ R3 (R)
e e
⇒ t.a = t.b = t.c = t.d = t.e = 0R , f or all a, b, c, d, e ∈ R
⇒ k|t
thus k = t.
Note: In the ring R3 (R), the multiplication of nonzero rhotrices A and B is equal to zero. Hence
R3 (R) has zero divisors and R3 (R) is not integral domain.
The following theorem characterize idempotent elements and nilpotent elements in a ring R3 (R).
Firstly we recall that definitions of idempotent and nilpotent elements in any ring. Let (R, +, .) be a
ring. An element a ∈ R is called idempotent if a2 = a and nilpotent if an = 0 for some positive integer
n.
Journal of New Theory 29 (2019) 32-41 / Rhotrices Ring 36
Theorem 2.5. Let R be a ring with identity 1R and c be an idempotent element in R. Then
* 0R +
0R c 0R is an idempotent element in R3 (R).
0R
* 0R +2 * 0R + * 0R +
Proof. 0R c 0R = 0R c2 0R = 0R c 0R . But all idempotents elements
0R 0R 0R
in the ring R3 (R) is not this form. For example; let R be a ring LK (E), where LK (E) is a Leavitt
Path Algebra [7] and
v2
e2
e1 v1
E: * +
e2
Let A = 0 v1 0 be in R3 (LK (E)). Since in a ring LK (E), v1 .v1 = v1 , v1 .v2 = v2 .v1 = 0,
0
e2 .v1 = v2 .e2 = 0, v1 .e2 = e2 .v2 = e2 , v1 .e1 = e1 .v1 = e1 , v2 .e1 = e1 .v2 = 0. Therefore,
* e2 .v1 + v1 .e2 + * e2 +
A2 = 0 v1 .v1 0 = 0 v1 0 =A
0 0
Theorem 2.6. Let R be a ring with identity 1R and c be a nilpotent element in the ring R. Then
* a +
b c d is a nilpotent element in a ring R3 (R).
e
* a +
Proof. We give any A = b c d ∈ R3 (R) and let be c a nilpotent element in a ring R.
e
Since c is a nilpotent element, there exits n ∈ Z+ such that cn = 0R . Then, h(An ) = cn = 0R and
A2n = An .An = 0R3 (R) .
In particularly; If R is a commutative ring. Then cn = 0R implies that An+1 = 0R3 (R) .
In this section, ideals of rhotrices ring have been investigated. Furthermore characterizations of
maximal ideals and prime ideals have been given.
* +
a
Theorem 3.1. Let R be a ring and I be an ideal of R. Then, M = b c d : c ∈ I is an
e
ideal in R3 (R).
* +
a
Proof. Since I is an ideal of R, M is a subset of R3 (R) and M 6= ∅. We give any A = b c d ,
e
* + * +
a1 x
B = b1 c1 d1 ∈ M , and C = y z t ∈ R3 (R). Then c, c1 ∈ I and z ∈ R,
e1 u
b
c + (−c1 ), z.c, c.z ∈ I . Hence A+(−B) ∈ M and A ⊙ C, C ⊙ A ∈ M . Thus, M is an ideal in R3 (R).
Theorem 3.2. Let R be a ring and R3 (R) be a ring of rhotrices.
Proof. (⇒) Let I be an ideal of R. Then I ⊆ R and I 6= ∅. Thus R3 (I) ⊆ R3 (R) and R3 (I) 6= ∅.
For any A ∈ R3 (I), since h(A) ∈ I, R3 (I) is an ideal in R3 (R) from Theorem 3.1.
(⇐) Let R3 (I) be an ideal of R3 (R). It is easy check that I 6= ∅, I ⊆ R.
We give any a, b ∈ I and r ∈ R .
* + * +
0R 0R
i. a ∈ I ⇒ A = 0R a 0R ∈ R3 (I) and b ∈ I ⇒ B = 0R b 0R ∈ R3 (I). Since
0R 0R
* +
0R
b
R3 (I) is an ideal of R3 (R), A+(−B) = 0R a − b 0R ∈ R3 (I) and a − b ∈ I
0R
* +
r
ii. r ∈ R ⇒ C = 0R 0R 0R ∈ R3 (R). Since R3 (I) is a ideal of R3 (R),
0R
* +
a.r
A⊙C = 0R 0R 0R ∈ R3 (I) and a.r ∈ I
0R
Similarly,
* +
r.a
C⊙A= 0R 0R 0R ∈ R3 (I) and r.a ∈ I
0R
Consequently, I is an ideal of R.
Corollary 3.3. Let K be a subset of R3 (R). K is an ideal in R3 (R) if and only if there exists an
ideal I in R such that h(A) ∈ I, for all A ∈ K.
Conversely, let R3 (I) be a principal ideal in R3 (R). Then there exist P ∈ R3 (R) such that
R3 (I) = (P ).*We will show that
+ I = (h(P )).
0R
a∈I⇒ 0R a 0R ∈ R3 (I) = (P )
0R
⇒ a = h(P ).x , x ∈ R ⇒ a ∈ (h(P )). Thus I ⊆ (h(P )). Since R3 (I) = (P ) , h(P ) ∈ I. Then
(h(P )) ⊆ I. Thus I = (h(P )).
Theorem 3.5. Let R be a ring and I be an ideal of R. Then,
* +
a+I
R3 (R/I) = b+I c+I d+I : a + I, b + I, c + I, d + I, e + I ∈ R/I
e+I
b and ” ⊙ ” and R3 (R)/R3 (I) isomorphic to ring R3 (R/I).
is a ring with as known operations ”+”
Proof. Since R/I is a ring, R3 (R/I) is a ring. We will show that R3 (R/I) ∼
= R3 (R)/R3 (I) . We
define f : R3 (R) → R3 (R/I) by
* +! * + * +
a a+I a
f b c d = b+I c+I d+I , for any b c d ∈ R3 (R)
e e+I e
* + * +
a x
It is easy to see that f is a well-defined. We give any A = b c d and B = y z t ∈
e u
R3 (R)
i. * +
(a + x) + I
b
f (A+B) = (b + y) + I (c + z) + I (d + t) + I
* (e + u) +
+I * +
a+I x+I
= b+I c+I d+I +b y+I z+I t+I
e+I u+I
b (B)
= f (A)+f
and * +!
a.z + c.x
f (A ⊙ B) = f b.z + c.y c.z d.z + c.t
* e.z + c.u +
(a.z + c.x) + I
= (b.z + c.y) + I (c.z) + I (d.z + c.t) + I
(e.z + c.u) + I
= f (A) ⊙ f (B)
ii.
* + * +!
a a
Kerf = b c d ∈ R3 (R) : f b c d = 0R3 (R/I)
e e
* +
a
= b c d : a, b, c, d, e ∈ I
e
= R3 (I)
Journal of New Theory 29 (2019) 32-41 / Rhotrices Ring 39
iii.
* +! * +
a a
Imf = f :
b c d b c d ∈ R3 (R)
e e
* +
a+I
= b+I c+I d+I : a, b, c, d, e ∈ R
e+I
= R3 (R/I)
Thus, f is a surjective.
Proof. Suppose that K is not a maximal ideal in R3 (R). Then there exists an ideal J in R3 (R)
such that K ⊆ J ⊆ R3 (R).
Since J is an ideal, there exists an ideal I in R such that h(A) ∈ I for arbitrary A ∈ J and since
K ⊆ J, M ⊆ I but I M because for every A ∈ J, h(A) ∈ / M . Therefore there exists an ideal I in
R. However, this gives a contradiction since M is a maximal ideal of R.
Theorem 3.9. Let R be a ring and R3 (P ) be a prime ideal of ring R3 (R). Then P is a prime ideal
of R.
Example 3.10. Although 3Z is a prime ideal in the ring Z, R3 (3Z) is not a prime ideal in the ring
R3 (Z). Indeed,
* + * +
−2 4
A⊙B = 5 3 1 ⊙ 1 6 −1
2 2
* +
0
= 33 18 3 ∈ R3 (3Z)
18
but A ∈
/ R3 (3Z) and B ∈
/ R3 (3Z).
Corollary 3.11. Let K be an ideal in R3 (R). K is a prime ideal in R3 (R) if and only if there exists
a prime ideal P in R such that h(A) ∈ P , for all A ∈ K.
Proof. Let K be a prime ideal in R3 (R). Then by Corollary 3.3, P is an ideal in R. We will show
that P is a prime.
a.R.b ⊆ P , for all a, b ∈ R. Then a.c.b ∈ P , for all c ∈ R . By hypothesis, there exists A ∈ K
such that h(A) = a.c.b ∈ P . There exists X, Y, Z rhotrices such that A = X ⊙ Y ⊙ Z and
h(X) = a, h(Y ) = b, h(Z) = c. Since K is a prime ideal in R3 (R) and A ∈ K, either X ∈ K or
Z ∈ K. Hence either a ∈ P or c ∈ P . Thus P is a prime ideal in R.
Conversely, let P be a prime ideal in R. Then by Corollary 3.3 , K is a ideal in R3 (R). Let
X ⊙ R3 (R) ⊙ Y ⊆ K, for any X, Y ∈ R3 (R). Then X ⊙ C ⊙ Y ∈ K, for all C ∈ R3 (R). Hence
h(X ⊙ C ⊙ Y ) = h(X).h(C).h(Y ) ∈ P and since P is a prime ideal in R, either h(X) ∈ P or h(Y ) ∈ P .
Thus either X ∈ K or Y ∈ K and so K is a prime ideal in R3 (R).
Example 3.12. Let R = Z6 = {0, 1, 2, 3, 4, 5}. Then, A = (0), B = (2), C = (3), D = Z6 are
ideals in Z6 . Hence, K = R3 (R) , K1 = 0R3 (R) , K2 = R3 (B) , K3 = R3 (C) ,
* + * + * +
R R R
K4 = R A R , K5 = R B R , and K6 = R C R
R R R
are ideals in R3 (Z6 ). Furthermore since B and C are prime ideals in Z6 , K5 ve K6 are prime ideals
in R3 (Z6 ). It is easy to see that K5 ve K6 are prime ideals in R3 (Z6 ).
=K ❈
a❈
④④④ ❈❈
④④ ❈❈
④ ❈❈
④④
KO 6 KO 5
K3 a❈ = K2
❈❈ ④④
❈❈ ④④
❈❈ ④④
❈ ④④
K1
Furthermore since B and C are maximal ideals in Z6 , K5 ve K6 are maximal ideals in R3 (Z6 ).
From above graphic, it is easy to see that K5 ve K6 are prime ideals in R3 (Z6 ).
Acknowledgement
This paper has been the granted by the Muğla Sıtkı Koçman University Research Projects Coordina-
tion Office through Project Grant Number: 17-223.
Journal of New Theory 29 (2019) 32-41 / Rhotrices Ring 41
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[7] G. Abrams, P. Ara, M. S. Molina, The Leavitt Path Algebra of a Graph, Lecture Notes in Math-
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