Geodesic
Almost Abelian rings
Communications in Mathematics, Tome 21 (2013) no. 1, pp. 15-30
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

A ring $R$ is defined to be left almost Abelian if $ae=0$ implies $aRe=0$ for $a\in N(R)$ and $e\in E(R)$, where $E(R)$ and $N(R)$ stand respectively for the set of idempotents and the set of nilpotents of $R$. Some characterizations and properties of such rings are included. It follows that if $R$ is a left almost Abelian ring, then $R$ is $\pi $-regular if and only if $N(R)$ is an ideal of $R$ and $R/N(R)$ is regular. Moreover it is proved that (1) $R$ is an Abelian ring if and only if $R$ is a left almost Abelian left idempotent reflexive ring. (2) $R$ is strongly regular if and only if $R$ is regular and left almost Abelian. (3) A left almost Abelian clean ring is an exchange ring. (4) For a left almost Abelian ring $R$, it is an exchange $(S,2)$ ring if and only if $\mathbb Z/2\mathbb Z$ is not a homomorphic image of $R$.
A ring $R$ is defined to be left almost Abelian if $ae=0$ implies $aRe=0$ for $a\in N(R)$ and $e\in E(R)$, where $E(R)$ and $N(R)$ stand respectively for the set of idempotents and the set of nilpotents of $R$. Some characterizations and properties of such rings are included. It follows that if $R$ is a left almost Abelian ring, then $R$ is $\pi $-regular if and only if $N(R)$ is an ideal of $R$ and $R/N(R)$ is regular. Moreover it is proved that (1) $R$ is an Abelian ring if and only if $R$ is a left almost Abelian left idempotent reflexive ring. (2) $R$ is strongly regular if and only if $R$ is regular and left almost Abelian. (3) A left almost Abelian clean ring is an exchange ring. (4) For a left almost Abelian ring $R$, it is an exchange $(S,2)$ ring if and only if $\mathbb Z/2\mathbb Z$ is not a homomorphic image of $R$.
Classification : 16A30, 16A50, 16D30, 16E50
Keywords: left almost Abelian rings; $\pi$-regular rings; Abelian rings; $(S, 2)$ rings 
@article{COMIM_2013_21_1_a1,
     author = {Wei, Junchao},
     title = {Almost {Abelian} rings},
     journal = {Communications in Mathematics},
     pages = {15--30},
     year = {2013},
     volume = {21},
     number = {1},
     mrnumber = {3067119},
     zbl = {06202722},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2013_21_1_a1/}
}
TY  - JOUR
AU  - Wei, Junchao
TI  - Almost Abelian rings
JO  - Communications in Mathematics
PY  - 2013
SP  - 15
EP  - 30
VL  - 21
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/COMIM_2013_21_1_a1/
LA  - en
ID  - COMIM_2013_21_1_a1
ER  - 
%0 Journal Article
%A Wei, Junchao
%T Almost Abelian rings
%J Communications in Mathematics
%D 2013
%P 15-30
%V 21
%N 1
%U http://geodesic.mathdoc.fr/item/COMIM_2013_21_1_a1/
%G en
%F COMIM_2013_21_1_a1
Wei, Junchao. Almost Abelian rings. Communications in Mathematics, Tome 21 (2013) no. 1, pp. 15-30. http://geodesic.mathdoc.fr/item/COMIM_2013_21_1_a1/

[1] Badawi, A.: On abelian $\pi$-regular rings. Comm. Algebra, 25, 4, 1997, 1009-1021, | DOI | MR | Zbl

[2] Camillo, V.P., Yu, H.P.: Exchange rings, Units and idempotents. Comm. Algebra, 22, 12, 1994, 4737-4749, | DOI | MR | Zbl

[3] Chen, H.Y.: A note on potent elements, Kyungpook. Math. J., 45, 2005, 519-526, | MR

[4] Chen, W.X.: On semiabelian $\pi$-regular rings. Intern. J. Math. Sci., 23, 2007, 1-10, | DOI | MR | Zbl

[5] Ehrlich, G.: Unit regular rings. Portugal. Math., 27, 1968, 209-212, | MR | Zbl

[6] Henriksen, M.: Two classes of rings that are generated by their units. J. Algebra, 31, 1974, 182-193, | DOI | MR

[7] Kim, N.K., Nam, S.B., Kim, J.Y.: On simple singular $GP$-injective modules. Comm. Algebra, 27, 5, 1999, 2087-2096, | DOI | MR | Zbl

[8] Lam, T.Y., Dugas, A.S.: Quasi-duo rings and stable range descent. J. Pure Appl. Algebra, 195, 2005, 243-259, | DOI | MR | Zbl

[9] Nicholson, W.K.: Lifting idempotents and exchange rings. Trans. Amer. Math. Soc., 229, 1977, 269-278, | DOI | MR | Zbl

[10] Nicholson, W.K.: Strongly clear rings and Fitting's Lemma. Comm. Algebra, 27, 8, 1999, 3583-3592, | DOI | MR

[11] Tuganbaev, A.: Rings close to regular. 2002, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 545, | MR | Zbl

[12] Vaserstein, L.N.: Bass' First stable range condition. J. Pure Appl. Algebra, 34, 1984, 319-330, | DOI | MR

[13] Wang, S.Q.: On op-idemotents. Kyungpook Math. J., 45, 2005, 171-175, | MR

[14] Warfield, R.B.: A krull-Schmidt theorem for infinite sums of modules. Proc. Amer. Math. Soc., 22, 1969, 460-465, | DOI | MR | Zbl

[15] Warfield, R.B.: Exchange rings and decompositions of modules. Math. Ann., 199, 1972, 31-36, | DOI | MR | Zbl

[16] Wei, J.C.: Certain rings whose simple singular modules are nil-injective. Turk. J. Math., 32, 2008, 393-408, | MR | Zbl

[17] Wei, J.C., Chen, J.H.: $Nil$-injective rings. Intern. Electr. Jour. Algebra, 2, 2007, 1-21, | MR | Zbl

[18] Wu, T., Chen, P.: On finitely generated projective modules and exchange rings. Algebra Coll., 9, 4, 2002, 433-444, | MR | Zbl

[19] Yu, H.P.: On quasi-duo rings. Glasgow Math. J., 37, 1995, 21-31, | DOI | MR | Zbl