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Let A be a regular local ring of dim A = 3 and p a prime ideal in A of dim A/p = 1. We put Rs(p) = (here t denotes an indeterminate over A) and call it the symbolic Rees algebra of p. With this notation the authors [5, 6] investigated the condition under which the A-algebra Rs(p) is Cohen-Macaulay and gave a criterion for Rs(p) to be a Gorenstein ring in terms of the elements f and g in Huneke’s condition [11, Theorem 3.1] of Rs(p) being Noetherian. They furthermore explored the prime ideals p = p(n1, n2, n3) in the formal power series ring A = k[X, Y, Z] over a field k defining space monomial curves and Z = with GCD(n1, n2, nz) = 1 and proved that Rs(p) are Gorenstein rings for certain prime ideals p = p(n1 n2, n3). In the present research, similarly as in [5, 6], we are interested in the ring-theoretic properties of Rs(p) mainly for p = p(n1 n2) nz) and the results of [5, 6] will play key roles in this paper.

There is given a characterization of Noetherian local ringsα d d ) holds true for all systems a 1 , a 2 , • • • , a d of parameters and integers n ≥ 1, where the suffix α runs over α = (α 1 , α 2 , • • • , α d ) ∈ Z d such that α i ≥ 1 for all 1 ≤ i ≤ d and d i=1 α i = d + n − 1.

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Abstract. Let Q = (a 1 , a 2 , · · · , as) ( A) be an ideal in a Noetherian local ring A. Then the sequence a 1 , a 2 , · · · , as is A-regular if every a i is a non-zerodivisor in A and if