i cant associate definitions of local-global rings,and also cant understand its exeamples.i am so confused,does anyone know about it?
defn1: A commutative ring is local-global,if every polynomial over R with finitely indeterminates,which represents units locally,represents units globally.
defn2:R be commutative ring.assume that every polynomial whose values generate unit ideal actually takes on an invertible values.
defn3:R is local-global if every primitive polynomial by values represent an inverse.
and important note:if R is local global then R quotient jakobson radical of R is local global (because they have same units)
SOME EXAMPLES OF L.G RINGS
**1.fields ** are trivial example.i can understand it.since localization of field is itself,if f represent unit locally then automaticly f represent unit globally.
2.semilocal rings i cant understand why?this rings have finite number of max ideal. and its jakobson radical quotient ring is semisimple.maybe someone has idea about it?i hope:)
3.rings with zero krull dimension if it is domain it is clear.becuse it is a field.but if it is a just ring,i have no idea.i can just say,in this rings prime ideal=max ideal.and in such rings every element is either unit or zero divisor
i wrote all i know and all i found.in books this axamples arent solved,they seems like they are trivial.but i reaally cant understand...