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"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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Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
1 Universytetska Str., 79000 Lviv, Ukraine
Tel: (+380 322) 394 172
E-mail: oromaniv at franko.lviv.ua

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