 Neha Patil (Editor)
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to have a positive mindset Arithmetical ringUpdated on Sign inIn algebra, a commutative ring R is said to be arithmetical (or arithmetic) if any of the following equivalent conditions holds: - The localization
R
m
of R at
m
is a uniserial ring for every maximal ideal
m
of R.
- For all ideals
a
,
b
, and
c
,
a
∩
(
b
+
c
)
=
(
a
∩
b
)
+
(
a
∩
c
)
- 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.
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