Nonassociative rings with some Jordan product
identities in the center
K. Madhusudhan Reddy
Department of Mathematics, School of Advanced Sciences,
VIT University, Vellore -632 014, Tamil Nadu.
CorrespondingAuthorE-mail:drkmsreddy@yahoo.in
ABSTRACT:
Quadri et al proved that if R is an associative
ring satisfying the identity(x
y)2
= x2y2
for all x, y in R, then R is commutative. Many results
have been proved for associative rings. This paper contains generalization of some
results on nonassociative rings with unity. Jordan product type identities were
takenin the center of nonassociative rings. Here xy
= xy + yx is the Jordan product. The following identities satisfies the commutativity
of a nonassociative ring with unity in the center.(i) (xy)
ÎU,
(ii) (xy)2
– (xy)
ÎU,(iii) (xy2)
– (x2y)
ÎU,(iv) (xy)2
– (x2y2)
ÎU,
(v) (x2y2)z2
– (xy)zÎU,(vi) (xy)2z2
– (xy)zÎU,(vii) (xy2)z
– (xy)zÎUand
(viii) (x2y2)z2
– (xy)zÎU for all x, y, z in R.
KEYWORDS: Nonassociative ring, center, and Char. ≠ n
AMS Classifications :17D05, 17D15.
INTRODUCTION:
Throughout this paper R represents nonassociative ring
with char. ¹ 2. The center U of R is defined as U = {uÎR / [u, R] = 0}. It is also called as
commutative center. A ring R is of characteristic ≠ n if nx = 0 implies x
= 0 for all x in R and n a natural number.
Theorem 1:Let R be a nonassociative ring of char. ¹ 2 with unity satisfying (xy)
ÎU for allx, y in R. Then R is commutative.
Proof : By hypothesis (xy)ÎU
or xy + yxÎU. (1)
Now we replace x with x + 1 in (1). Then (x + 1)y +
y(x + 1) ÎU.
or xy + yx + 2yÎU. (2)
Using (1) in (2), we get
2yÎU. Since
R is of char. ¹ 2, we get yÎU.
i.e., xy = yx, for all x in R.
Hence R is commutative.
Theorem 2 :Let R be a nonassociative ring of char. ¹ 2 with unity satisfying
(xy)2
– (xy)
ÎUfor all x, y in R. Then R is commutative.
Proof : By hypothesis (xy)2
– (xy)
ÎU
or (xy + yx)2 – (xy
+ yx) ÎU. (3)
Now by replacing x with x +1 in (3), we
get
((x+1)y + y(x+1))2 –
((x+1)y + y(x+1)) ÎU
i.e., (xy + yx)2 + (2y)2
+ (xy + yx) (2y) + (2y) (xy + yx) – (xy + yx+
2y) ÎU.(4)
Using (3) in (4), we get
4y2 + (xy + yx) (2y)
+ (2y) (xy + yx) – 2yÎU.
(5)
Since R is of char. ¹ 2, we get
2y2+ (xy + yx) (y) + (y)
(xy + yx)– yÎU. (6)
Now by replacing x with x + 1 in (6), we
get
2y2 + (xy + yx+2y) (y)
+ (y) (xy + yx + 2y)– yÎU.
(7)
Using (6) in (7), we get
4y2ÎU.
Since R is of char. ¹ 2, we get
y2ÎU.
(8)
By replacing y with y + 1 in (8) and
using (8), we get
2yÎU. Since R is of char. ¹ 2, we have yÎU
orxy = yx for all x in R.
Hence R is commutative.
Theorem 3 :Let R be a nonassociative ring of char. ¹ 2 with unity satisfying
(xy2)
– (x2y)
ÎUfor all x, y in R. Then R is commutative.
Proof : By hypothesis (xy2)
– (x2y)
ÎU
or (xy2 + y2x)
– (x2y + yx2) ÎU.
(9)
Now by replacing x with x + 1 in (9), we
get
(xy2 + y2x+2y2)
– (x2y + yx2+2y2)ÎU. (10)
Using (9) in (10), we get
4y2ÎU.
Since R is of char. ¹ 2, we get
y2ÎU.
(11)
By replacing y with y + 1 in (11) and
using (11), we get
2yÎU. Since R is of char. ¹ 2, we have yÎU
orxy = yx for all x in R.
Theorem 4 : Let R be a nonassociative ring of char. ¹ 2 with unity satisfying
(xy2)
– (x2y)
ÎU for all x, y in R. Then R is commutative.
Proof : By hypothesis (xy2)
– (x2y)
ÎU,
or xy2 + y2x – x2y
– yx2ÎU. (12)
By replacing x with x + 1 in (12), we get
(x + 1)y2 + y2(x + 1) – (x + 1)2y
– y(x + 1)2ÎU
or xy2 + y2x – x2y
– yx2 – 2xy – 2yx – 2yÎU. (13)
Now by using (12) and (13), we get
2xy + 2yx + 2yÎU. (14)
Since R is of char. ¹ 2, we get
xy + yx+ yÎU. (15)
Again by replacing x with x + 1 in (15) and using (15),
we obtain
2yÎU. Since
R is of char. ¹ 2, we get yÎU or xy = yx, for all x in R.
Hence R is commutative.
Theorem 5 : Let R be a nonassociative ring of char. ¹ 2 with unity satisfying
(x2y2)z2
– (xy)zÎU for all x, y, z in R. Then R is
commutative.
Proof : By hypothesis (x2y2)z2
– (xy)zÎU. (16)
Now we replace z with z + 1 in (16)Then
(x2y2)
(z + 1)2– (xy)
(z + 1) ÎU
or (x2y2)z2
– (xy)z+
2(x2y2)z
+ (x2y2)
– (xy)
ÎU. (17)
Using (16) in (17), we obtain
2(x2y2)z+
(x2y2)
– (xy)
ÎU. (18)
Again by replacing z with z + 1 in (18),
we get
2(x2y2)(z
+ 1) + (x2y2)
– (xy)
ÎU
or 2(x2y2)z
+ (x2y2)
– (xy)
+ 2(x2y2)
ÎU. (19)
Using (18) in (19) and R is of char. ¹ 2, we get
x2y2
ÎU
or x2y2
+ y2x2ÎU. (20)
By replacing x with x + 1 in (20) and
using (20), we get
2xy2 + 2y2x
+ 2y2ÎU. Since R is of char. ¹ 2, we obtain
xy2 + y2x
+ y2ÎU. (21)
Again by replacing x with x + 1 in (21)
and using (21), we get 2y2ÎU.
Since R is of char. ¹ 2, we get y2ÎU. (22)
By replacing y with y + 1 in (22) and
using (22), we get
2yÎU. Since R is of char. ¹ 2, we have yÎU
orxy = yx for all x in R.
Hence R is commutative.
Theorem 6: Let R be a nonassociative ring of char. ¹ 2 with unity satisfying
(xy)2z
– (xy)zÎU for all x, y, z in R. Then R is commutative.
Proof : By hypothesis (xy)2z–
(xy)zÎU. (23)
Now we replace z with z + 1 in (23) Then
(xy)2
(z + 1) – (xy)
(z + 1) ÎU
or (xy)2z
+ (xy)z-
(xy)z
– (xy)
ÎU.
(24)
Using (23) in (24), we obtain
(xy)2–
(xy)
ÎU.
(25)
From theorem 2 the results follows.
Hence it is commutative.
Theorem 7: Let R be a nonassociative ring of char. ¹ 2 with unity satisfying
(xy2)
z – (xy)zÎU for all x, y, z in R. Then R is
commutative.
Proof : By hypothesis (xy2)
z– (xy)zÎU.
(26)
Now we replace z with z + 1 in (26) Then
(xy2)
(z + 1) – (xy)
(z + 1) ÎU
or (xy2)z
+ (xy2)z-
(xy)
z – (xy)
ÎU.
(27)
Using (26) in (27), we obtain
(xy2)
– (xy)
ÎU.
Or (xy2+y2x) – (xy +
yx)
(28)
By replacing x with x + 1 in (12) and
using (12), we get
2y2 – 2yÎU. (29)
Since R is of char. ¹ 2,y2 – yÎU. (30)
By replacing y with y + 1 in (30) and
using (30), we get
2yÎU. (31)
Since R is of char. ¹ 2, we have yÎU
orxy = yx for all x in R.
REFERENCES:
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(1992–93), 39–42.
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Univ. of Chicago press, London (1969).
3. Quadri, M.A., Khan, M.A. and Ashraf, M.
“Some elementary theorems for rings”, Math. Stud., 56 (1988) 223–226.
4. Ram Awtar. “On the commutativity of
nonassociative rings”, Publicationes Mathematicae, 22 (1975) 177–185.