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

Irwing Kaplansky

# Arithmetical properties of principal ideals in morphic rings

Olexiy Sorokin

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

Abstract:

Theorem 1. If R is a morphic ring and aR ~ bR, cR ~ dR then
a) aR+cR ~ bR∩dR;
b) aR∩cR ~ bR+dR;
c) acR ~ (b:c)=(d:a);
d) tensor product of the principal ideals aR and bR is morphic to their interstion.

Theorem 2. If R is a morphic ring and aR ~ bR then a and b locally behave like usual integers.

Theorem 3. Morphic reduced ring is a von Neumann regular ring.

Theorem 4. If R is a commutative ring and I is its ideal then R/I is morphic iff for any element a+I of R/I there is some b+I in R/I such that (I:a)=bR+I,  (I:b)=aR+I.

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