"I liked the algebraic way of looking at things.
I’m additionally fascinated when the algebraic method is applied to infinite objects”.

Irwing Kaplansky

# A morphic ring of neat range 1

Oksana Pihura

Department of Algebra and Logic

Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv

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.

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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O. Pihura, 30 Sep, 2016

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