Neha Patil

Arithmetical ring

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In algebra, a commutative ring R is said to be arithmetical (or arithmetic) if any of the following equivalent conditions holds:

  1. The localization R m of R at m is a uniserial ring for every maximal ideal m of R.
  2. For all ideals a , b , and c , a ( b + c ) = ( a b ) + ( a c )
  3. For all ideals a , b , and c , a + ( b c ) = ( a + b ) ( a + c )

The last two conditions both say that the lattice of all ideals of R is distributive.

An arithmetical domain is the same thing as a Prüfer domain.

References

Arithmetical ring Wikipedia


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