Ore condition
The ( left or right ) Ore - conditions are in the ring theory, a branch of algebra, a criterion which allows us to generalize the formation of quotient bodies or general localizations to the case in which the underlying ring is not commutative is. They are named after their discoverer, Øystein Ore. Rings that they meet are called ( left or right ) Ore -rings.
- 3.1 Ring of (legal) quotient
- 3.2 Properties and examples
Motivation
In commutative algebra the localization of rings is a useful tool. This is, roughly speaking, is that elements of a subset of the ring are made invertible or " approved as denominator ". For this to be meaningful defined, there is only necessary that the amount is multiplicative and one contains (usually is also still required ).
As soon as one tries to generalize this approach to non-commutative rings, one encounters several problems. Although you can always abstract form a ring in which the elements of are invertible and to that in the commutative case satisfies an appropriate universal property analogous, but this generally has poor properties and is not easy to specify in concrete terms. Difficulties arise even for divisors of zero rings. For example, it has been shown that there are divisors of zero rings, which can be embedded in any division ring. In particular, it can not in full generality a kind of " quotient skew field " give analogous to the quotient field of integral domains.
The Norwegian mathematician Øystein Ore was in 1931 in an article a criterion that allows the formation of certain rings of quotients. Ores considerations were later generalized by Keizo Asano and others.
Special case: zero divisor free rings
Be a ring (with 1 ) without zero divisors. meets the right Ore condition if for all:
.
That is, and have other common multiple " right " except the 0 is then also called a right Ore ring.
Similarly, the left Ore condition defined by for all.
Formation of " quotient skew fields "
Complies with the right Ore condition, then one can similarly form a quotient skew field as in the formation of the quotient field. The elements are again written as fractions, such as
With.
Here are two " breaks " and identifies if there are other elements, and so applies. ( Formally, thus defines an equivalence relation on the set, and denotes the equivalence class of. )
For these " breaks " the addition and the multiplication of certain formulas are now defined, which are a little more complicated than the usual rules at break arithmetic. For the definitions (like that above identification was indeed an equivalence relation ) in each case the right Ore condition must be exploited.
The so- defined addition and multiplication make the amount of those " breaks " actually a skew field, and the mapping defines an embedding from to.
In addition the following universal property holds: If a ring homomorphism such that for all a unit is in, so too is clearly proceeds to a ring homomorphism.
Analogously, define everything " from the left ". It should be noted that a ring may fulfill the left Ore condition without being a right- Ore ring, and vice versa ( see examples). Is a ring, however, both a left-and a right Ore ring ( one says simply " Ore -Ring"), the corresponding left or right quotient division ring are isomorphic.
Properties and examples
- Everyone ( links-/rechts- ) Noetherian ring without divisors of zero satisfying ( Links-/Rechts- ) Ore condition.
- A zero-divisor- free ring is exactly then a ( Links-/Rechts- ) Ore - ring if it is uniform as ( Links-/Rechts- ) module over itself, ie two non-trivial submodules have non-trivial intersection.
- The ring of integer quaternions is an Ore ring and has a quotient division ring of the rational quaternions.
- Be and the Frobeniushomomorphismus (ie ). Then the ring of Schiefpolynome is a zero-divisor free left Ore ring, but not a right Ore ring.
Ore -rings
Now let an arbitrary non - commutative ring. It can be left or right zero divisor occur, and these may initially not even be reasonably accepted as the denominator. As denominator instead provides the set of all regular elements ( ie, those that are neither left nor right zero divisor ) to. This is multiplicative containing one but not the 0 in the above was a special case.
Meets the right Ore condition if exist for all elements, so that
Or equivalent:
.
(One can easily show that this is in the above special case is equivalent to the condition given there. )
A ring that satisfies the right Ore condition is called right Ore ring. By overturning of all products we obtain the analogue definitions for the left Ore condition and left Ore rings.
Ring of (legal) quotient
We now want a ring of right quotients and an injective homomorphism construct, which should fulfill the following conditions:
- For all one unit.
- Every element of can be described as writing with suitable.
Again, analogous definitions " from the left " possible to write then.
The set of Ore indicates a precise criterion for when there is such a ring of quotients:
Called also the "classical ring of right quotients " and is denoted by. (Analog everything " from the left " with the label. )
If both a left-and a right Ore ring is, the corresponding classical rings of left and right quotients are isomorphic.
Properties and examples
- Every commutative ring is an Ore ring. ( All Links-/Rechts-Eigenschaften coincide, and the common localization is the ring of quotients. )
- Be a body, the polynomial ring in the variable and the ring of rational functions in. Then the ring is
Although a right Ore ring with classical ring of right quotients, but R is not a left Ore ring. For example, that is, the left Ore condition is violated.
Further generalization
The above definition of a ring ( right ) ratio can be easily modified to more general transfer (as opposed to the "classic " regular elements of ). In general, however, we can not ask for more then that is injective. A reasonable substitute therefore the additional condition:
- Ker.
It turns out that such a ring can be accurately then formed by ( right ) quotient with respect if the following properties satisfied:
- For all exist elements such that. (This is only the generalization of right Ore condition for the amount. )
- Be. Is there a with, so even one with. ( This condition was previously empty, since only consisted of regular elements. )