Geodesic
A morphic ring of neat range one
Algebra and discrete mathematics, Tome 20 (2015) no. 2, pp. 325-329.

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We show that a commutative ring $R$ has neat range one if and only if every unit modulo principal ideal of a ring lifts to a neat element. We also show that a commutative morphic ring $R$ has a neat range one if and only if for any elements $a, b \in R$ such that $aR=bR$ there exist neat elements $s, t \in R$ such that $bs=c$, $ct=b$. Examples of morphic rings of neat range one are given.
Keywords: Bezout ring, neat ring, clear ring, elementary divisor ring, stable range one, neat range one. 
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O. Pihura; B. Zabavsky. A morphic ring of neat range one. Algebra and discrete mathematics, Tome 20 (2015) no. 2, pp. 325-329. http://geodesic.mathdoc.fr/item/ADM_2015_20_2_a9/

[1] Bass H., Algebraic K-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968 | MR | Zbl

[2] Henriksen M., “Some remarks about elementary divsor rings II”, Michigan Math. J., 3 (1955), 159–163 | DOI | MR

[3] Houston E., Taylor J., “Arithmetic properties in pulbacks”, J. Algebra, 310 (2007), 235–260 | DOI | MR | Zbl

[4] Kaplansky I., “Elementary divisirs and modules”, Trans. Amer. Math. Soc., 66 (1949), 464–491 | DOI | MR | Zbl

[5] McGovern W. Wm., “Bezout rings with almost stable range 1”, J. of Pure and Appl. Algebra, 24 (1982), 25–40 | DOI | MR

[6] Nicholson W. K., “Lifting idempotents and exchange rings”, Trans. Amer. Math. Soc., 229 (1977), 269–278 | DOI | MR | Zbl

[7] Nicholson W. K., Sanchez Campos E., “Rings with the dual of the isomorphism theorem”, J. Algebra, 271 (2004), 391–406 | DOI | MR | Zbl

[8] Roitman M., “The Kaplansky condition and rings of almost stable range 1”, Trans. Amer. Math. Soc., 141 (2013), 3013–3019 | MR

[9] Zabavsky B. V., “Diagonal reduction of matrices over finite stable range”, Mat. Stud., 41 (2014), 101–108 | MR | Zbl

[10] Zabavsky B. V., Diagonal reduction of matrices over rings, Mathematical Studies, Monograph Series, XVI, VNTL Publishers, Lviv, 2012, 251 pp. | MR | Zbl