# local-global rings and its examples

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...

## 1 Answer

I found about examples of local global rings and wanted to share.. Fields are trivially local global ,since localization of field is itself.and similarly it holds for local rings since localization of local ring is again itself . For semilocal rings:Let R be semilocal ring with n maximal ideals.If we define naturel map from R to direct sum of modulo max ideals of R,we see that this map is clearly onto.and kernel of this map is intersection of all max ideals,namely jakobson radical of R. By first isomorphisim theorem modulo the j(R)of R isomorphic to direct sum of modulo the max ideals of R,which are fields. Finally we reach that R quotient Rad(R) is isomorphic to direct sum of fields.and we have fields are local global.Direct sum of a local global rings are again local global,hence semilocal ring R is local-global. For rings with krull-dimension is zero: i can say that in such rings every element is either zero divisor or unit.i think that**(but not yet sure)**,set of all zero divisors of R only max ideal of R,since other element are units and there can not be any unit in any proper ideal.ıf i am right rings with krull zero are local ring.and we know that local rings are local global.