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Internat. J. Math. & Math. Sci. (1988) 9-14 VOL. ii NO. TWO PROPERTIES OF THE POWER SERIES RING H. AL-EZEH Department of Mathematics University of Jordan Amman, Jordan (Received July 31, 1986 and in revised form October 29, 1986) For a commutative ring with unity, A, it is proved that the power series ring ABSTRACT. A X is a PF-ring if and only if for any two countable subsets S and T of A such that S Sann(T), there exists c e ann(T) such that bc A b for all b e S. Also it is proved A that a power series ring A X is a PP-ring if and only if A is a PP-ring in which every increasing chain of idempotents in A has a supremum which is an Idempotent. KEY WORDS AND PHRASES. Power series ring, PP-ring, PF-ring, flat, projective, annihilator ideal and idempotent element. 1980 AMS SUBJECT CLASSIFICATION CODE. 13B. i. INTRODUCTION. Rings considered in this paper are all commutative with unity. power series ring over the ring A. principal ideal is a flat A-module. Let A X be the Recall that a ring A is called a PF-ring if every Also a ring A is called a PP-ring if every principal ideal is a projective A-module. It is proved in Ai-Ezeh [I] that a ring A is a PF-ring if and only if the annihilator of each element a e A, ann(a), A exists c e ann(a) such that bc b. is a pure ideal, that is for all b e ann(a) there A A ring A is a PP-ring if and only if for each a e A, A In Brewer [3], semihereditary A power series rings over von Neumann regular rings are characterized. In this paper we ann(a) is generated by an idempotent, see Evans [2]. characterize PF- power series rings and PP- power series rings over arbitrary rings. For any reduced ring A (i.e. a ring with no nonzero nilpotent elements), it was proved in Brewer et al. [4] that ann A (a X0 + alX + ...) N X]] where N is the annihilator of the ideal generated by the coefficients a O, 2. MAIN RESULTS. al,... H. AL-EZEH I0 Any PF-ring A is a reduced ring. LEMMA I. Assume that there is a nonzero nilpotent element in A. Let n be the least n Because A is a PF0. So a e such that a positive integer greater than A nn-I 0 a (ab) a. Thus a such that ab ring there exists be ann A n-i 0. since ba Contradiction. So any PP-ring is a reduced ring. PROOF. ann(an-l). n-lbn-I (an-I THEOREM 2. The power series ring A {bo, countable sets S b I, b X 2 exists c e ann(T) such that b.c is a PF-ring if and only if for any two {a 0, and T al, ...} ann(T), there such that S A O, i, 2 b.i for i 1 A PROOF. First, we prove that A X is a PF-ring. Let g(X) b + + and 0 b2X + f(X) a g(X) ann 0 alX + and let (f(X)). Then g(X) f(X) 0. AX The ring A is inparticular a PF-ring because for all b e ann(a), there exists A c e ann(a) such that bc A b.a. I S So by Lemma b. O, for all i ..; I, A is a reduced ring. O, j Thus 2, So {b0, b ann(ao, A ). a So by assumption, there exists ann(a 0, a b for all i such that b c Hence g(X)c O, g(X) I’ i i A and c e ann (f(X)). Consequently the ring A X is a PF-ring. Conversely, assume c e AEX A [IX]] is a PF-ring. b Let _c {b0, Then g(x) f(X) O. ann(a0,A ...) al, + blX + + al+ ........ Let g(X) Therefore g(X) e ann b (f(X)). and f(X) 0 a Thus there exists h(X) Co+ClX+... AEX g(X). (f(X)) such that g(X) h(X) in ann 0 AEx Consequently, h(X) f(X) c.a. and bic So c o 0 0, for all i for all j Z i. 0 e ann(a O, a I, I) 0 and g(X) (h(X) ...) and j {Co, c b.1 for Hence bic 0 0. Since 0, i, 2 A is reduced, anx(a 0, all i O, I, 0 bi(c0-1) and a and for all i O. bi(c0-1) Therefore the above condition holds. Because any PP-ring is a PF-ring, every PP-ring is a reduced ring. ring A, a partial order relation can be defined by a s b if ab a 2. On a reduced The following lemma is given in Brewer[3] and Brewer et al.[4]. LEMMA 3. PROOF. ab a 2 defined above on a reduced ring A is a partial order. The relation Clearly the relation s is reflexive. and ba b 2 So a-b 2 a 2 2ab + b2 Now assume a O. b and b a. Because A is reduced a Then b O, TWO PROPERTIES OF THE POWER SERIES RING or a To prove transitivity of E, assume a b. b and b Ii So ab c. a 2 and bc b 2 Consider (ac ab) 2 2 a (c 2 2 =a (c 2cb + b 2 b 2 2 b)(c + b) ab(c 0 b) because b(c Since A is reduced, ac 0. THEOREM 4. The power series ring AX 0 or ac ab ab a 2 Therefore a < b. is a PP-ring if and only if A is a PP-ring in which every increasing chain of idempotents of A with respect to N has a supremum which is an idempotent element in A. Assume A PROOF. idempotents in A X X Let a e A. is a PP-ring. eA [[ (a) are in A, ann X AX 0, rea ea (a). b e ann for all r e A. 0 ann(a). A Hence eA eg(X) for some g(X) Thus b b X A That is b e eA. 0 + chain of idempotents in A, (e ann AEx Since eei So e. N e.1 for all i + (e AEx 0 y is a PP-ring and eA. Because We claim ann(a) A Now let b e ann(a). Hence A eb Consequently, b + O. blX X X i 0 0e < e be an increasing 2 is a PP-ring and since idempotents of A Now we claim e sup{e O, e 2 e e i i }. Let y be an upper bound of {e 0, e O, e iX + ...). for some c e A. ec Consequently, ce)(l (I e) y(l e ec e) + ec i- e So e y. Therefore e ...}. sup{e O, e To prove the other way around, consider a%xf(X)) Hence ann (f(X)) AEX an(a O, a X ...}. O, I, y e ann Thus + e) e eA Because A Ao e X ei(l y for i Hence + 0 0, 1 X Whence A is a PP-ring. To complete the proof of this direction, let e are in Since A ann(a 0 A =/ i=O a ann(a i) )X where f(X) a 0 + alX + So 12 H. AL-EZEH e.A, e. e. i=0 because A is a PP-ring. Let do= e 0, d d e0e d n e n-I n One can easily check that f e.A= i=0 d. A i=O Also it is clear that o _> d > d i- d o < I- d 2 Therefore _< d 1- d 2 By assumption, this increasing chain of idempotents has a supremum which is an idempotent. Let Sup{l (I d di) o d I, d d d 2, ...} d. 0, fo,:all i i We claim that Now d > d d)d. So (i i. (I -d)A Thus (I d)A i=0 diA. Hence d. diA for all i 0, diA- i=O Let y d)A. (i diA Then y i,0,1, diY i, Consequently (i di)(l + y) diY l-d Because Therefore yd i d d 2 2. diYi diYi 1 i y y" for all i 0, i d i y So TWO PROPERTIES OF THE POWER SERIES RING Because Sup{l d $ d Hence That is 0. dy y g (I d }, So d I, d(l y) d Thus y(l d) y yd d y. d)A. O 2 dy y d.A Therefore 13 (i d)A. i;O Consequently, (f(X)) A[[X]] ann Therefore A[[X]] (I- d) A[[X]] is a PP-ring. REFERENCES i. AL-EZEH, H. On Some properties of Polynomial rings. I.J.M.M.S To appear Pac. J. Math. 41(1972) 687-697. 2. EVANS, M. 3. BREWER, J. 4. RUTTER, E. and WATKINS, J. Coherence and weak global dimension of BREWER, J. R [[X]] when R is Von Neumann regular, J. of Algebra 46(1977) 278-289. On commutative PP-rings. series over commutative rings". Lecture Notes in pure and applied Mathematics No. 64, Marcel Dekker, New York and Basel (1981). "Power