On Centralizing and Strong Commutativity Preserving Maps of Semiprime Rings
We study some properties of centralizing and strong commutativity preserving maps of semiprime rings.
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| author | Gölbaşı, Ö. Huang, Shuliang Koç, E. Голбаши, О. Хуанг, Шулян Коч, Е. |
| author_facet | Gölbaşı, Ö. Huang, Shuliang Koç, E. Голбаши, О. Хуанг, Шулян Коч, Е. |
| author_sort | Gölbaşı, Ö. |
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| datestamp_date | 2019-12-05T09:47:54Z |
| description | We study some properties of centralizing and strong commutativity preserving maps of semiprime rings. |
| first_indexed | 2026-03-24T02:16:23Z |
| format | Article |
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UDC 512.5
Shuliang Huang (Chuzhou Univ., China),
Ö. Gölbaşı, E. Koç (Cumhuriyet Univ., Turkey)
ON CENTRALIZING AND STRONG COMMUTATIVITY
PRESERVING MAPS OF SEMIPRIME RINGS*
ПРО ЦЕНТРАЛIЗУЮЧI ТА СИЛЬНI ВIДОБРАЖЕННЯ НАПIВПРОСТИХ
КIЛЕЦЬ, ЩО ЗБЕРIГАЮТЬ КОМУТАТИВНIСТЬ
We study some properties of centralizing and strong commutativity preserving maps of semiprime rings.
Вивчаються деякi властивостi централiзуючих та сильних вiдображень напiвпростих кiлець, що зберiгають комута-
тивнiсть.
1. Introduction. Let R will be an associative ring with center Z. For any x, y ∈ R, the symbol [x, y]
stands for the commutator xy − yx and the symbol xoy denotes for the anticommutator xy + yx.
Recall that a ring R is prime if xRy = 0 implies x = 0 or y = 0 and R is semiprime if xRx = 0
implies x = 0. A prime ring obviously semiprime. An additive mapping d from R into itself is
called derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R. Let S be a nonempty subset of R. A
mapping F from R to R is called centralizing on S if [F (x), x] ∈ Z, for all x ∈ S and is called
commuting on S if [F (x), x] = 0, for all x ∈ S. Also, F is called commutativity preserving on a
subset S of R if [x, y] = 0 implies [F (x), F (y)] = 0, for all x, y ∈ S. The mapping F is called strong
commutativity preserving (simply, SCP) on S if [x, y] = [F (x), F (y)], for all x, y ∈ S. The study
of centralizing and commuting mappings was initiated by Posner in [14]. Over the last fifteen years,
several authors have proved commutativity theorems for prime rings or semiprime rings admitting
automorphisms or derivations which are centralizing and commuting on appropriate subsets of R (see,
e.g., [4, 6, 11, 12] and references therein). On the other hand, there is also a growing literature SCP
maps and derivations. For more information on SCP maps and derivations, we refer to [5, 7, 10].
In [8], M. A. Chaudhry and A. B. Thaheem showed that if R is a semiprime ring and f is an
endomorphism of R, g is an epimorphism of R such that [f (x) , g (x)] ∈ Z, then [f (x) , g (x)] = 0
holds for all x ∈ R. In [1], A. Ali, M. Yasen and M. Anwar showed that if R is a semiprime ring,
f is an endomorphism which is a strong commutativity preserving map on a nonzero ideal U of R,
then f is commuting on U . In [13], M. S. Samman proved that an epimorphism of a semiprime ring
is strong commutativity preserving if and only if it is centralizing.
Morever, in [2], M. Asraf and N. Rehman showed that a prime ring R with a nonzero ideal I
must be commutative if it admits a derivation d satisfying d(xy)± xy ∈ Z, for all x, y ∈ R. In [3],
the authors explored this result for a generalized derivation of R.
In this paper, we prove some results of centralizing and strong commutativity preserving maps
of semiprime rings. In Theorem 1, we extend a result of M. A. Chaudhry and A. B. Thaheem [8]
(Theorem 2.2). In Theorem 2 is an analogues of [1] (Theorem 1) and Theorem 4 is an extension of
* The first author is supported by the Anhui Provincial Natural Science Foundation (1408085QA08) and the Key
University Science Research Project of Anhui Province (KJ2014A183) and also the Training Program of Chuzhou University
(2014PY06) of China.
c© SHULIANG HUANG, Ö. GÖLBAŞI, E. KOÇ, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2 279
280 SHULIANG HUANG, Ö. GÖLBAŞI, E. KOÇ
[3] (Theorem 2.5). Also, we shall make use of the following basic identities without any specific
mention:
i) [x, yz] = y[x, z] + [x, y]z,
ii) [xy, z] = [x, z]y + x[y, z],
iii) xyoz = (xoz)y + x[y, z] = x(yoz)− [x, z]y,
iv) xoyz = y(xoz) + [x, y]z = (xoy)z + y[z, x].
2. Results.
Lemma 1 ([9], Lemma 1). Let R be a semiprime ring and U a nonzero ideal of R. If z in R
centralizes the set [U,U ] , then z centralizes U.
Theorem 1. Let R be a semiprime ring with charR 6= 2, f and g be two endomorphisms of R
and U is a nonzero right ideal of R. If [f (u) , g (u)] ∈ Z for all u ∈ U, then [f (u) , g (u)] = 0 for
all u ∈ U.
Proof. A linearization of [f (u) , g (u)] ∈ Z yields that
[f (u) , g (v)] + [f (v) , g (u)] ∈ Z for all u, v ∈ U.
Replacing v by u2 in this equation, we get
[f (u) , g (u)] g (u) + g (u) [f (u) , g (u)] + f (u) [f (u) , g (u)] + [f (u) , g (u)] f (u) ∈ Z.
Using the hypothesis and charR 6= 2, we obtain that
g (u) [f (u) , g (u)] + f (u) [f (u) , g (u)] ∈ Z for all u ∈ U.
Commuting this term with f (u) , we arrive at
[f (u) , g (u)]2 = 0 for all u ∈ U.
Since the center of a semiprime ring contains no nonzero nilpotent elements, we conclude that
[f (u) , g (u)] = 0 for all u ∈ U.
Theorem 1 is proved.
In particular, if we take g = I, where I : R→ R is an identity endomorphism, then we have the
following corollary which is a generalization of [4] (Lemma 2) for the case when characteristic is
different from two.
Corollary 1. Let R be a semiprime ring with charR 6= 2, f be an endomorphism of R and U is
a nonzero right ideal of R. If f is centralizing on U, then f is commuting on U.
Theorem 2. Let R be a semiprime ring, f and g be two endomorphisms of R and U is a nonzero
ideal of R. If [f (u) , g (v)] = [u, v] for all u, v ∈ U, then g is commuting on U.
Proof. By the hyphothesis, we have
[f (u) , g (v)] = [u, v] for all u, v ∈ U.
Substituting vw, w ∈ U for v in the above equation, we obtain that
[f (u) , g (v)] g (w) + g (v) [f (u) , g (w)] = [u, v]w + v [u,w] .
Using the hyphothesis, we arrive at
[u, v] g (w) + g (v) [u,w] = [u, v]w + v [u,w] ,
and so
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
ON CENTRALIZING AND STRONG COMMUTATIVITY PRESERVING MAPS OF SEMIPRIME RINGS 281
[u, v] (g (w)− w) + (g (v)− v) [u,w] = 0 for all u, v, w ∈ U. (1)
Replacing w by u in (1), we get
[u, v] (g (u)− u) = 0 for all u, v ∈ U.
Taking v by rv, r ∈ R in the last equation and using this equation, we see that
[u, r] v (g (u)− u) = 0 for all u, v ∈ U, r ∈ R,
and so
[u, r]RU (g (u)− u) = 0 for all u ∈ U, r ∈ R.
Now, we let ℘ = {Pα |α ∈ Λ} be a family of prime ideals with ∩Pα = (0) . If P is a typical member
of ℘ and u ∈ U, then the last equation shows that
[u,R] ⊆ P or U (g (u)− u) ⊆ P.
Suppose that ∃v ∈ U such that [v,R] * P. Thus U (g (v)− v) ⊆ P. Let w is any element of U
such that [v + w,R] ⊆ P . Hence [w,R] * P. Indeed, if [w,R] ⊆ P, then [v,R] ⊆ P. It contradicts
[v,R] * P. Therefore we get [w,R] * P. This implies that U (g (w)− w) ⊆ P for all w ∈ U.
If [v + w,R] * P, then U (g (v + w)− (v + w)) ⊆ P for all w ∈ U, and so U (g (w)− w) ⊆ P
for all w ∈ U. Hence we obtain that U (g (w)− w) ⊆ P for all w ∈ U, for any cases. Therefore
[U,U ] (g (w)− w) ⊆ P for all w ∈ U.
Since ∩Pα = (0) , we have
[U,U ] (g (w)− w) = (0) for all w ∈ U. (2)
On the other hand, taking u instead of v in (1), we obtain
(g (u)− u) [u,w] = 0 for all u,w ∈ U.
Again appliying similar arguments as above, we get
(g (w)− w) [U,U ] = (0) for all w ∈ U. (3)
Using (2) and (3), we conclude that (g (w)− w) ∈ CR ([U,U ]) for all w ∈ U. By Lemma 1, we
obtain that (g (w)− w) ∈ CR (U) for all w ∈ U. Thus [g (w)− w,w] = 0 for all w ∈ U. This implies
that [g (w) , w] = 0 for all w ∈ U, and so g is commuting on U.
Theorem 2 is proved.
If we have f = g, then we can give the following corollary which is a generalization of [1]
(Theorem 1).
Corollary 2. Let R be a semiprime ring, f be an endomorphism of R and U is a nonzero ideal
of R. If f is strong commutativity preserving on U, then f is commuting on U.
Corollary 3. Let R be a semiprime ring, f be an endomorphism of R and U is a nonzero ideal
of R. If f satisfies one of the following conditions:
(i) f(uv) = uv for all u, v ∈ U,
(ii) f(uv) = −uv for all u, v ∈ U,
(iii) for each u, v ∈ U , either f(uv) = uv or f(uv) = −uv,
then f is commuting on U .
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
282 SHULIANG HUANG, Ö. GÖLBAŞI, E. KOÇ
Proof. (i) By the hypothesis, we get f(uv) = uv for all u, v ∈ U. Thus, we have
f (uv − vu) = f (uv)− f (vu) = uv − vu.
Therefore [f (u) , f (v)] = [u, v] , for all u, v ∈ U. By Corollary 2, we arrive at f is commuting on U.
(ii) Using the same arguments in the proof of (i), we find the required result.
(iii) For each u ∈ U , we put Uu = {v ∈ U | f(uv) = uv} and U∗
u = {v ∈ U | f(uv) = −uv}.
Then (U,+) = Uu∪U∗
u . But a group cannot be the union of proper subgroups. Hence we get U = Uu
or U = U∗
u . By the same method in (i) or (ii), we complete the proof.
Theorem 3. Let R be a semiprime ring, f and g be two endomorphisms of R and U is a nonzero
ideal of R. If f (u) g (v)− uv = 0 for all u, v ∈ U, then g is commuting on U.
Proof. By the hypothesis, we have f (u) g (v) = uv for all u, v ∈ U. Replacing v by vw, we
find that
f(u)g(v)g(w) = uvw for all u, v ∈ U.
Using the hypothesis in this equation, we get uvg(w) = uvw, and so uv(g(w) − w) = 0. This can
be written as U2(g(w)− w) = (0) and implies that
[U,U ](g(w)− w) = 0 for all w ∈W. (4)
Substituting uw for u in the hypothesis and using this, we find that f(u)wv = uwv. Taking
g(t)w instead of w in this equation, we get
f(u)g(t)wv = ug(t)wv,
and so
utwv = ug(t)wv for all u, v, w, t ∈ U.
The above expression implies that u(g(t) − t)U2 = (0). Replacing u by ur, r ∈ R in this equation,
we obtain that
uR(g(t)− t)U2 = (0) for all u, t ∈ U.
Now as in the proof of Theorem 2, we let ℘ = {Pα |α ∈ Λ} be a family of prime ideals with
∩Pα = (0) . If P is a typical member of ℘ and u ∈ U, then the last equation shows that
(g(t)− t)U2 = (0) for all t ∈ U,
and so
(g(t)− t)[U,U ] = (0) for all t ∈ U. (5)
Using (4) and (5), we conclude that (g (t)− t) ∈ CR ([U,U ]) for all t ∈ U. By Lemma 1, we
obtain that (g (t)− t) ∈ CR (U) for all t ∈ U. Thus [g (t)− t, t] = 0 for all t ∈ U. This implies that
[g (t) , t] = 0 for all t ∈ U, and so g is commuting on U.
Theorem 3 is proved.
Theorem 4. Let R be a semiprime ring, f and g be two endomorphisms of R and U is a nonzero
ideal of R. If f (u) g (v)− uv ∈ Z for all u, v ∈ U, then g is commuting on U.
Proof. Replacing v by vw, w ∈ U in the hypothesis, we get
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
ON CENTRALIZING AND STRONG COMMUTATIVITY PRESERVING MAPS OF SEMIPRIME RINGS 283
f (u) g (v) g (w)− uvw ∈ Z for all u, v, w ∈ U.
Let f (u) g (v)− uv = α, α ∈ Z. Writing f (u) g (v) = α+ uv in the above equation, we have
(α+ uv) g (w)− uvw ∈ Z,
and so
uv (g (w)− w) + αg (w) ∈ Z for all u, v, w ∈ U. (6)
Commuting this term with g (w), we arrive at
−uv [w, g (w)] + u [v, g (w)] (g (w)− w) + [u, g (w)] v (g (w)− w) = 0.
Substituting ru, r ∈ R for u in the last equation, we obtain that
−ruv [w, g (w)]+ru [v, g (w)] (g (w)− w)+r [u, g (w)] v (g (w)− w)+[r, g (w)]uv (g (w)− w) = 0.
That is
[r, g (w)]uv (g (w)− w) = 0,
and so
[r, g (w)]RU2 (g (w)− w) = 0 for all w ∈ U, r ∈ R.
Let ℘ = {Pα |α ∈ Λ} be a family of prime ideals with ∩Pα = (0) , P a typical member of ℘ and
u ∈ U. For each w ∈ U either U2 (g (w)− w) ⊆ P or [r, g (w)] ∈ P for all r ∈ R. We assume
that ∃v ∈ U such that [r, g (v)] /∈ P. Let u is any element of U such that [r, g (u+ v)] ∈ P . This
implies that [r, g (u)] /∈ P. Thus U2 (g (u)− u) ⊆ P for all u ∈ U. If [r, g (u+ v)] /∈ P, then
U2 (g (u+ v)− (u+ v)) ⊆ P for all u ∈ U, and so U2 (g (u)− u) ⊆ P for all u ∈ U. Hence we
get U2 (g (u)− u) ⊆ P for all u ∈ U.
Since P arbitrary and ∩Pα = (0) , we arrive at
U2 (g (u)− u) = (0) for all u ∈ U. (7)
That is
[U,U ] (g (u)− u) = (0) for all u ∈ U. (8)
Multipliying (7) on the left by (g(u)− u), we have
(g(u)− u)U2 (g (u)− u) = (0) for all u ∈ U.
Again multipliying this equation on the right by U2, we obtain that
(g(u)− u)U2 (g (u)− u)U2 = (0) for all u ∈ U,
and so
(g(u)− u)U2R (g (u)− u)U2 = (0) for all u ∈ U.
By the semipimeness of R, we conclude that
(g (u)− u)U2 = (0) for all u ∈ U,
and so
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
284 SHULIANG HUANG, Ö. GÖLBAŞI, E. KOÇ
(g (u)− u) [U,U ] = (0) for all u ∈ U. (9)
By (8) and (9), one easily checks that (g(u)− u) ∈ CR ([U,U ]), for all u ∈ U. By Lemma 1, we get
(g (u)− u) ∈ CR (U) for all u ∈ U. Thus [g (u)− u, u] = 0 for all t ∈ U, and so g is commuting onU.
Theorem 4 is proved.
We can give a following corollary which is a generalization of Corollary 3 (i).
Corollary 4. Let R be a semiprime ring, f be an endomorphism of R and U is a nonzero ideal
of R. If f (uv)− uv ∈ Z for all u, v ∈ U, then f is commuting on U.
Theorem 5. Let R be a semiprime ring, f be an endomorphism, g an epimorphism of R and U
is a nonzero ideal of R. If f (u) og (v) = uov for all u, v ∈ U, then g is commuting on U.
Proof. Writing v by vr, r ∈ R in the hypothesis, we have
(f(u)og(v))g(r) + g(v)[g(r), f(u)] = (uov)r + v[r, u] for all u, v ∈ U, r ∈ R.
Using the hypothesis, we obtain that
(uov) (g (r)− r) + g (v) [g (r) , f (u)] = v [r, u] for all u, v ∈ U, r ∈ R. (10)
Replacing v by uv in (10), we get
u (uov) (g (r)− r) + [u, u] v (g (r)− r) + g (u) g (v) [g (r) , f (u)] = uv [r, u] ,
and so
u ((uov) (g (r)− r)− v [r, u]) + g (u) g (v) [g (r) , f (u)] = 0 for all u, v ∈ U, r ∈ R.
Using (10) in the above equation, we see that
−ug (v) [g (r) , f (u)] + g (u) g (v) [g (r) , f (u)] = 0.
That is
(g (u)− u) g (v) [g (r) , f (u)] = 0 for all u, v ∈ U, r ∈ R.
Since g is onto, we arrive at
(g (u)− u)V [r, f (u)] = 0 for all u ∈ U, r ∈ R,
where V is an ideal of R. Thus (g (u)− u)V R [r, f (u)] = 0, for all u ∈ U. By the semiprimeness
of R, it must contain a family ℘ = {Pα |α ∈ Λ} of prime ideals such that ∩Pα = (0) . Let
P denote a fixed one of the Pα. From the last equation, it follows that for each u ∈ U either
(g (u)− u)V ⊆ P or [r, f (u)] ⊆ P for all r ∈ R. Assume that ∃v ∈ U such that [r, f (v)] * P.
Therefore (g (v)− v)V ⊆ P .
Suppose w is any element of U . If [r, f (v + w)] ∈ P , then [r, f (w)] /∈ P. Indeed, if
[r, f (w)] ∈ P , then [r, f (v)] ∈ P . It contradicts [r, f (v)] /∈ P. Hence we get [r, f (w)] /∈ P .
That is (g (w)− w)V ⊆ P for all w ∈ U. On the other hand, if [r, f (v + w)] /∈ P, then
(g (v + w)− (v + w))V ⊆ P. This implies that (g (w)− w)V ⊆ P for all w ∈ U. For any cases
(g (w)− w)V ⊆ P for all w ∈ U and so (g (w)− w) [V, V ] ⊆ P for all w ∈ U. Since P is arbitrary
and ∩Pα = (0) , we obtain that
(g(w)− w)[V, V ] = (0) for all w ∈ U. (11)
Taking rv instead of v, r ∈ R in the hypothesis, we find that
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
ON CENTRALIZING AND STRONG COMMUTATIVITY PRESERVING MAPS OF SEMIPRIME RINGS 285
f(u)og(r)g(v) = uorv,
g(r)(f(u)og(v)) + [f(u), g(r)]g(v) = r(uov) + [u, r]v,
and so
(g(r)− r)(uov) + [f(u), g(r)]g(v) = [u, r]v for all u, v ∈ U, r ∈ R. (12)
Replacing v by vu in the last equation, we get
((g(r)− r)(uov)− [u, r]v)u+ [f(u), g(r)]g(v)g(u) = 0.
Using (12) in the above equation, we arrive at
[f(u), g(r)]g(v)(g(u)− u) = 0 for all u, v ∈ U, r ∈ R.
Since g is onto, we have
[f(u), r]V (g(u)− u) = 0 for all u ∈ U, r ∈ R,
where V is an ideal of R. Using similar arguments as above, we can prove that
[V, V ](g(w)− w) = (0) for all w ∈ U. (13)
By equations (11) and (13), one easily checks that (g (w)− w) ∈ CR ([V, V ]) for all w ∈ U.
By Lemma 1, we obtain that (g (w)− w) ∈ CR (V ) for all w ∈ U. Since V = g(U), we have
(g (w)− w) ∈ CR (g(U)) for all w ∈ U. Thus [g (w)− w, g(w)] = 0 for all w ∈ U. This implies
that [g (w) , w] = 0 for all w ∈ U, and so g is commuting on U.
Theorem 5 is proved.
Corollary 5. Let R be a semiprime ring, f be an epimorphism of R and U is a nonzero ideal of
R. If f (uov) = uov for all u, v ∈ U, then f is commuting on U.
1. Ali A., Yasen M., Anwar M. Strong commutativity preserving mappings on semiprime rings // Bull. Korean Math.
Soc. – 2006. – 43, № 4. – P. 711 – 713.
2. Ashraf M., Rehman N. On derivations and commutativity in prime rings // East-West J. Math. – 2001. – 3, № 1. –
P. 87 – 91.
3. Ashraf M., Asma A., Shakir A. Some commutativity theorems for rings with generalized derivations // Southeast
Asain Bull. Math. – 2007. – 31. – P. 415 – 421.
4. Bell H. E., Martindale W. S. Centralizing mappings of semiprime rings // Can. Math. Bull. – 1987. – 30. – P. 92 – 101.
5. Bell H. E., Daif M. N. On commutativity and strong commutativity preserving maps // Can. Math. Bull. – 1994. –
37, № 4. – P. 443 – 447.
6. Bresar M. Centralizing mappings and derivations in prime rings // J. Algebra. – 1993. – 156. – P. 385 – 394.
7. Bresar M. Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings // Trans.
Amer. Math. Soc. – 1993. – 335, № 2. – P. 525 – 546.
8. Chaudhry M. A., Thaheem A. B. Centralizing mappings and derivations on semiprime rings // Demonstr. Math. –
2004. – 37, № 2. – P. 285 – 292.
9. Daif M. N., Bell H. E. Remarks on derivations on semiprime rings // Int. J. Math. and Math. Sci. – 1992. – 15, № 1. –
P. 205 – 206.
10. Ma J., Xu X. W. Strong commutativity-preserving generalized derivations on semi prime rings // Acta Math. Sinica
(English Ser.). – 2008. – 24, № 11. – P. 1835 – 1842.
11. Mayne J. H. Centralizing automorphisms of prime rings // Can. Math. Bull. – 1976. – 19. – P. 113 – 115.
12. Thaheem A. B., Samman M. S. Centralizing mappings on semiprime rings // Int. J. Pure and Appl. Math. – 2002. –
3. – P. 249 – 254.
13. Samman M. S. On strong commutativity-preserving maps // Int. J. Math. and Math. Sci. – 2005. – 6. – P. 917 – 923.
14. Posner E. C. Derivations in prime rings // Proc. Amer. Math. Soc. – 1957. – 8. – P. 1093 – 1100.
Received 26.04.13
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 2
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| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d3/e510d5bca22fcc940de3026ef766c6d3.pdf |
| spelling | umjimathkievua-article-19802019-12-05T09:47:54Z On Centralizing and Strong Commutativity Preserving Maps of Semiprime Rings Про централізуючі та сильні вiдображення напiвпростих кілець, що зберiгають комутативнiсть Gölbaşı, Ö. Huang, Shuliang Koç, E. Голбаши, О. Хуанг, Шулян Коч, Е. We study some properties of centralizing and strong commutativity preserving maps of semiprime rings. Вивчаються дєякі властивості централiзуючих та сильних відображень напiвпростих кілєць, що зберігають комутативність. Institute of Mathematics, NAS of Ukraine 2015-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1980 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 2 (2015); 279-285 Український математичний журнал; Том 67 № 2 (2015); 279-285 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1980/980 https://umj.imath.kiev.ua/index.php/umj/article/view/1980/981 Copyright (c) 2015 Gölbaşı Ö.; Huang Shuliang; Koç E. |
| spellingShingle | Gölbaşı, Ö. Huang, Shuliang Koç, E. Голбаши, О. Хуанг, Шулян Коч, Е. On Centralizing and Strong Commutativity Preserving Maps of Semiprime Rings |
| title | On Centralizing and Strong Commutativity Preserving Maps of Semiprime Rings |
| title_alt | Про централізуючі та сильні вiдображення напiвпростих кілець, що зберiгають комутативнiсть |
| title_full | On Centralizing and Strong Commutativity Preserving Maps of Semiprime Rings |
| title_fullStr | On Centralizing and Strong Commutativity Preserving Maps of Semiprime Rings |
| title_full_unstemmed | On Centralizing and Strong Commutativity Preserving Maps of Semiprime Rings |
| title_short | On Centralizing and Strong Commutativity Preserving Maps of Semiprime Rings |
| title_sort | on centralizing and strong commutativity preserving maps of semiprime rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1980 |
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