Introduction. S. Abhyankar, W. Heinzer and P. Eakίn treated the following problem in [1]; if A[X]=B[Y], when is A isomorphic or identical to 5? Replacing the polynomial ring by the torus extension we shall take up the following problem; if A[X, X^]=B[Y, F" 1 ], when is A isomorphic or identical to B ? We say that A is torus invariant (resp. strongly torus The rolles played by polynomial rings in [1] are played by the graded rings in our theory. A graded ring A=^A i9 i^Z y with the property that

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... ^4 t φ0 for each iEZ, will be called a ^-graded ring. Main results are the followings. An afϊine domain A of dimension one over a field k is always torus invariant. Moreover A is not strongly torus invariant if and only if A has a graded ring structure. An affine domain of dimension two is not always torus invariant. We shall construct an affine domain of dimension two which is not torus invariant. Let A be an affine domain over k of dimension two. Assume that the field k contains all roots of *'unity" and is of characteristic zero. If A is not torus invariant, then A is a ^-graded ring such that there exist invertible elements of non-zero degree. In Section 1 we study elementary properties of graded rings. Especially we are interested in ^-graded rings with invertible elements of non-zero degree. In Section 2 we discuss some conditions for A to be torus invariant. In Section 3 we give several sufficient conditions for an integral domain to be strongly torus invariant. Some relevant results will be found in S. Iitaka and T. Fujita [2] . Section 4 is devoted to the proof of the main results mentioned above. In Section 5 we fix an integral domain D and we treat only Z)-algebras and Z)-isomorphisms there. We shall prove the following two results. When A is a D-algebra of tr. deg D A=\ and A is not Z)-torus invariant, A is a Z-graded ring such that D is contained in A o . If A is a iΓ-graded ring such as D=A Oy then the number of elements of the set of {Z)-isomorphic classes of Z)-algebras where d is the smallest positive integer among the degrees of units in A and Φ is the Euler function. Γd like to express my sincere gratitude to the referee for his valiable advices.