# A morphic ring of neat range 1

- Written by O. Pihura
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Department of Algebra and Logic* Faculty of Mechanics and Mathematics** Ivan Franko National University of L'viv*

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Abstract:

*A ring R is a commutative ring with nonzero identity. *

*All necessary definitions and facts can be found in [1], [2]. *

*An element a∈R\0 is called neat if R/aR is a clean ring. If all elements in R are neat then R called neat ring. *

*A commutative ring R is said to be of neat range 1 if for any a,b∈R such that aR+bR=R there exists t ∈ R such that for the element a+bt=c the ring R/cR is clean ring. *

*We say that a unit u modulo principal ideal aR, namely (ux-1)∈aR; x∈R, lifts to the neat element if there exists a neat element t∈R such that u-t∈aR. *

**Theorem 1.*** For a ring R the following statements are equivalent: *

*1) a neat range of a ring R is 1; **2) any unit modulo principal ideal of a ring R lifts to neat element. *

**Theorem 2.*** A morphic ring is a ring of neat range 1 if and only if for any pair of elements a,b∈R such that aR=bR there are neat elements s,t∈R such that as=b and b=at. *

**Theorem 3.*** Let R be an elementary divisor ring, then R is a ring of neat range one. *

**Theorem 4.*** Let R be an elementary divisor domain and a ∈ R\{0}. Then the factor-ring R/aR is a morphic ring of neat range one. *

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**References **

[1] Zabavsky B.V. Diagonal reduction of matrices over rings. Mathematical Studies, Monograph Series, V.XVI, VNTL Publishers, 2012, Lviv. 251 p.

[2] Zabavsky B.V. Diagonal reduction of matrices over finite stable range rings, Mat. Stud., 41 (2014), 101–108.

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