Partial Theorem List (
preliminary
)
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
[email protected]
Webpage
http://www.math.ufl.edu/
∼
squash/
5 December, 2011
(at
00:07
)
In this document,
ring
means “commutative ring”.
In the ring (
Z
,
+
,
·
,
0
,
1), an element
u
∈
Z
is a
unit
if
∃
w
∈
Z
st.
u
·
w
= 1.
(
This
w
is the
“
multiplicative
inverse
”
of
u
, is unique, and is often written
u
1
.
) In
Z
the
units are
±
1. But in
Z
12
, the ring of integers mod12,
the set of units is Φ(12), i.e,
{±
1
,
±
5
}
. In the ring
Q
of rationals,
each
nonzero element is a unit.
As a
last example, in the ring
G
:=
Z
+
i
Z
of
Gaussian
integers
, the units group is
{±
1
,
±
i
}
. (
This units group
is cyclic.
)
A
z
∈
Z
is a
zerodivisor
(
abbrev.
ZD
) if there
exists a nonzero
w
∈
Z
for which
zw
= 0. Every ring
has the
“
trivial zerodivisor
”
—zero itself. The ring
of integers doesn’t have others. In contrast, the non
trivial zerodivisors of
Z
12
comprise
{±
2
,
±
3
,
±
4
,
6
}
.
Exer. E1
: In an arbitrary ring Γ, the set ZD(Γ) is
disjoint
from Units(Γ).
Exer. E2
: Ring
Z
N
has no irreducible elements (
see
next
), since
Z
N
is the disjointunion ZD
t
Units.
Irreducible
and
prime
elements.
An
ele
ment
p
∈
Z
is
irreducible
/
prime
if it is
neither
a unit nor a zerodivisor,
and
:
IRRED:
For each factorization
p
=
b
·
c
, either
b
is
a unit or
c
is a unit.
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 Fall '07
 JURY
 Math, Calculus, Prime number, Integral domain, Ring theory, thm, Thm. SOTS Thm

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