ON EQUIVALENCE OF GRADED RINGS

Let R=⊕g∈GRg 
 be a G-graded ring. In this paper we define the homogeneousequivalence 
concept between graded rings. We discuss some properties of the G-graded rings and 
investigate which of these are preserved under homogeneous-equivalence maps. Furthermore, we give 
some results in graded ring theory and also some applications of this concept to Z-graded rings.


INTRODUCTION
Let G be a group with identity e.Then a ring R is said to be G-gradedrmg if there exist additive subgroups R9 of R such that R R and RRh C_ Rat for all g, h E G. We consider supp (R) {g E G" R 0}.The elements of R are called homogeneous of degree g.If z E R, then x can be written uniquely as xa where xa is the component of x in R. Also, we write h(R) U R.
gEG In this paper we define an equivalence relation on the set of all graded rings and give some applications of this relation.In Section 1, we define the homogeneous-equivalence concept between graded rings and give the necessary and sufficient conditions for two graded rings to be homogeneouslyequivalent.In Section 2, we discuss the relation between this new concept and some related concepts given in [1 In Section 3, we give some properties of graded rings and see which of these are preserved under homogeneous-equivalence maps.In Section 4, we give some useful results and applications of this new concept to Z-graded rings.We give the necessary and sufficient conditions for two first strongly Z-graded rings to be homogeneously-equivalent.

HOMOGENEOUS-EQUIVALENCE OF GRADUATIONS
In this section we define the homogeneous-equivalence concept between graded rings and give some of its properties.Also, we give the necessary and sufficient conditions for two graded rings to be homogeneously-equivalent.DEFINITION 1.1.Let G, H be groups, R be a G-graded ring and S be an H-graded ring.We say that R is homogeneously eqmvalent (shortly "h.e.") to S if there exists a ring isomorphism f-R S sending h(R) onto h(S).We call such an f a homogeneous-equivalence of R with S.
From now on G, H are _tn'oups.R is a G-graded ring and S is an H-graded ring unless otherwtse indacated.
"The author is on leave from Yarmouk University, Jordan.REMARK 1.2.The relation h.e. is an equivalence relation.PROPOSITION 1.3.An isomorphism f:R S of rings is a homogeneous equivalence if and only if there is a bijection b ofsupp (R) onto supp (S) such that f(R,) Sb(,) for all g E supp(R) In that case the bijection b is uniquely determined by the isomorphism f If R = 0, then the bijection b sends the identity element ec E supp(R) of G onto the identity element eH supp(S) of H. PROOF.Suppose f R S is a homogeneous equivalence map.Define b supp(R) supp(S) by b(g) h where f(R,) Sh.Then clearly b is a well-defined bijective map.The converse is obvious Suppose R # 0. To show that b(ec) =eH it is enough to show f(P) Sell.Since eH H, there exists g G such that f(R,) Ss and then f(R,)f(R,) SH, i.e., Ss C_ f(Rd).Therefore, f(R,2) S,s.But f is 1-1 implies Rg R,2 =# 0. Hence g g2 and then g ec, i.e., f(P) Let I be an ideal of a graded ring R. Then I is a graded ideal of R if I (Rg C I). PROPOSITION 1.4.Suppose R is h.e. to ,5' by f.Then I is a graded ideal of R if and only ill(I) is a graded ideal of S.
PROOF.An ideal I in a G-graded ring R is graded if and only if it is generated as an ideal by its subset I C h(R).Obviously this property is preserved under homogeneous equivalence maps.
REMARK 1.5.It follows from Proposition 1.4 that any property definable solely in terms of graded ideals of R must be preserved under homogeneous equivalence maps.
PROPOSITION 1.6.Let R be h.e. to S by f and I be a graded ideal of R. Then R/I is h.e. to S/f().PROOF.By Proposition 1.4, f(I) is a graded ideal of S and then S/f(I) is an H-graded ring Define o R/I sir(I) by o(r + I) f(r) + f(I).Then clearly is a ring isomorphism.Let h supp(S/f(I)).Then (S/f(I)) h 0 and hence there exists 8h Sh-f(I), i.e., h supp(S) Since R is h.e. to S by f, there exists g supp(R) such that f(R) S by Proposition 1.3 cII.((/z)) (s/l(z)).

DEFINITION 2.1 ([ I]
).We say that R is almost eqmvalent (shortly "a.e.") to S if there exists a ring isomorphism f R S such that for each h H, there exists g G with f(P) Sh.If R is a.e. to S and S is a.e. to R then we say R is equivalent to S. PROPOSITION 2.2.If R is a.e. to S then R is h.e. to S.
PROOF.Follows from Proposition 1.3 However, the converse of this proposition need not be true in general as we see in the following example.
EXAMPLE 2.3.Let K be a field and R S K[x] is the polynomial ring over K in one variable x. Let G 2a. Then R is a G-graded ring with R ( kx a,+ k E K, r 0, I, ...} for j a.Let H 20. Then S is an H-graded ring with So {:': K, o, ,...} S2 {kx:+ k _ g,r 0,1,...} Sa={kx"+2:keg, r=O,l,...} and Sa=0 otherwise. Clearly R is h.e. to S. IfR is a.e. to S by J' then there exists g E G with f(Rg) $3 0, e, Rg 0 a contradiction.PROPOSITION 2.4.R is a.e to ,5' if and only if the following two conditions are satisfied: (i) R is h.e. to S. (ii) If H-supp (S) =/= 0 then G-supp (R) #-0 PROOF.Suppose R is a.e. to S by f.Then (i) follows from Proposition 2.2 Assume H-supp (S) 0. Then there exists h E H such that Sh 0. Since R is a.e to S there exists g G such that f(R) 0. Thus R 0, i.e., G-supp (R) # 0.
Conversely, suppose R is h.e. to S by f.Let h H.If h supp(S) then by Proposition 1.3, there exists g G such that f(R)= Sh.If h supp(S) then by (ii) there exists g supp(R) So, 0 f(Rg) S, i.e., R is a.e to S by f.COROLLARY 2.5.R is equivalent to S if and only if the following two conditions are satisfied (i) R is h.e. to S.
PROOF.Similar to the proof of Proposition 2.3 in ].

PROPERTIES PRESERVED UNDER HOMOGENEOUS-EQUIVALENCE MAPS
In this section we give some properties of graded rings and see which of these are preserved under homogeneous-equivalence maps.For more details about the properties one can look in 1,2,3].DEFINITION 3.1.For a G-graded ring R we say