On nonsingular P-injective rings

Publicacions Matemàtiques, Vol 38 (1994), 455-461 . ON NONSINGULAR P -INJECTIVE RING S YASUYUKI HIRAN o Dedicated to the memory of Professor Hisao Tominag a A bstract  A ring R is said to be left p -injective if, for any principal left idea l 1 of R, any left R -homomorphism 1 into R extends to one of R into itself . In this note left nonsingular left p -injective rings ar e characterized using their maximal left rings of quotients and th e structure of semiprime left pinjective rings of bounded index i s investigated . A left R - module M is said to be p -injective if given any principa l left ideal 1 and any R-homomorphism o- : 1 —> M, there exists a n R-homomorphism cr : R --~ M that extends a . This notion was firs t introduced by Ikeda and Nakayama [81 . They proved that a ring R i s left p -injective if and only if every principal right ideal of R is a righ t annihilator . In [11, Proposition 1], it was proved that a ring R withou t nonzero nilpotent elements is von Neumann regular if and only if R i s left p --injective . However, in general, a semiprime left pinjective ring R need not be von Neumann regular . In this note, we give a characterization of a left nonsingular left p -injective ring using its maximal lef t ring of quotients and consider the structure of semiprime left pinjectiv e rings of bounded index . We also construct a semiprime left and righ t p-injective PI-ring which is not von Neumann regular and a semiprim e left p-injective PI-ring which is not right p -injective . For a subset F of a ring R, r R (F) (resp . Z R (F)} denote the righ t (resp . left) annihilator of F in R . To state our theorem, we need th e following definition . Definition 1 . Let R be a ring, and M aleft R-module . A submodul e P of M is said to be R -pure if aM n .P = aP for all a E R . 456  Y . HIRAN O Theorem 1 . Let R be a leftnonsingular ring and let Q denote th e maximal left quotient ring of R . Then the following statements are eqivalent : 1) R is left p-injective . 2) R R is R-pure in R Q . Rroof 1) 2) . Let a E R . By {4, Corollary 2 .31], is a v o n Neuman n regular ring . Hence, there exists an idempotent e E Q such that aQ = eQ . Then 1 R (a) = 1 Q (a) n R = l Q (eQ)n R = Q(1 — e) n R . By [8 , Theorem 1], the p -injectivity of R R implies that every principal righ t ideal of R is a right annihilator ideal . Hence aR = r R I R (a) = r R (Q (1 -- e) n R) D eQ n R = aQ n R D aR . This proves aQ n R = aR for al l aER . 2) ~ 1) . Let a E R . Then there exists an idempotent e E Q such tha t aQ = eQ . First we claim that r R (Q (1 — e) n R) = eQ n R . Clearly w e have that r R (Q (1 — e) n R) D eQ n R . To prove the converse inclusion , let b E r R (Q (1 — e) n R) . Since Q is the maximal left ring of quotient s of R, there exists an essential left ideal 1 of R such that 1(1  e) Ç R . Then 1(1 — e)b = 0 . Since R Q is nonsingular by [4, Proposition 2 .321 , this implies (1 — e) b = 0 . Therefore the converse inclusion also holds . Then r R l R (a) = r R (Q (1 — e) n R) = eQ n R = aQ n R = aR, becaus e R R is R -pure in R Q . By [8, Theorem 1], this implies the pinjectivity o f RR . ■ A ring R is said to be of bounded index (of nilpotence) if there is a positive integer n such that a' = 0 for each nilpotent element a of R . If n is the least such integer we say R has index n . For example, i t is well known that any semiprime ring satisfying a polynomial identit y is of bounded index ([3, Theorem 10 .8 .2I) . Recall 'that R is said to b e 7rregular if for each element a of R, there exists a positive integer m and an element x of R such that a m = a m xa m . On the other hand, R is said to be strongly 7rregular if far each element a of R, there exist s a positive integer k such that a~ R = a k+1 R . By [2, Théoréme 1] thi s definition is left-right symmetric, and hence such a ring is 7 rregular . In particular, every nonnil one-sided ideal of a strongly 7rregular rin g contains a nonzero idempotent . Proposition 1 . Let R be a semiprime p -injective ring of bounde d index . Then we have : (1) R is a strongly 7 r-regular ring . (2) The maximal left quotient ring Q of R is a finite direct product o f matriz rings over strongly regular self-injective rings . ON NONSINGULAR p-INJECTIVE RINGS  457 Proof : Let R is of index n and let a E R . Then ZR (an ) = lR (a n+l ) by [5, Proposition 2] . Hence we have a n R = rRIR (an ) = r R ZR (an+1 ) = 0 +1 R . This proves that R is a strongly 7 rregular ring . Since R is a semiprime ring of bounded index, by virtue of [G, Lemma 1 .11, ever y nonzero one-sided ideal of R contains a nonzero idempotent . Hence, th e assertion (2) follows from [5, Theorem 9 and (2) in Remarks] . ■ Assume that R is a left p -injective ring without nonzero nilpoten t elements . Then, by Proposition 1, R is a strongly 7 rregular ring o f index 1, that is, R is a strongly regular ring . Hence we obtain [11 , Proposition 11 . Also we have the followin g Corollary 1 . Let R be a semiprime p-injective ring of bounded index . Then R is von Neumann regular if and only if the union of any chain o f semiprime ideals of R is a semiprime ideal . In consequence, a semiprim e p-injective ring R which is finitely generated as a moduleover its cente r is von .ZlTeumann regular . Proof : By Proposition 1, R is strongly nregular, and so every prim e factor ring of R is regular by [T, Proposition 2] . Now the result follow s from [3, Theorem 1 .11 . If R is finitely genereted over its center, then R satisfies a polynomial identity, and hence R is of bounded index . Also , by the proof of [1, Theorem 11, we know that the union of any chain o f semiprime ideals of R is a semiprime ideal of R . ■ Now we shall generalize the construction technique of semiprime ring s used in [101 . Definition 2 . Let S be a ring and let T be a subring of S . For a n infinite set 1, (SIT)' denotes the subring of the direct product S I of I' s copies of S consisting of all s = (s i ), for which s i E T for all but a finit e number of i E 1 . (SIT) ( ' ) denotes the subring of S I consisting of al l s = (si ), for which s i = t for some t E T for all but a finite number o f i E I . Lemma 1 . Let S be a semisimple Artinian ring, and let 1 be a n infinite set . Let R be a subring of S I containing S ( I ) , the direct sum o f I's copies of S . Then R is a nonsingular semiprime ring and its maxima l left quotient ring Q is S . Proof : Let a = S (I) . Clearly a is an ideal of both R and S I . It i s easy to see that bab D for any nonzero b E R . Hence R is a semiprim e ring . Clearly R is of bounded index . Hence, by [5, Proposition 4], th e semiprime ring R of bounded index is a left (and right) nonsingular ring . 458  Y . HIRAN O Let K be a nonzero submodule of R SI . Then we can easily see tha t o aK C K n R . Hence RSI is an essential extension of R R . Clearly S I is a regular, left self-injective ring . Hence, by [4, Proposition 2 .11 and Corollary 2 .311, we conclude that Q = S I . n Proposition 2 . Let S be a semisimple Artinian ring, let T be a sub - ring of S and let 1 be an infinite set . Then the following statements ar e equivalent : 1)(SIT)' is a left p-injective ring . 2) (S I T) ( I ) is a left p-injective ring . 3) T T is T-pure in T S . Rroof : 1) <=> . 3) . Let R = (SIT)' . By Lemma 1, the maximal lef t ring Q of quotients of R is S I . Assume first that TT is T-pure in T S . Let (a) E R and let (c i ) E (a)Q n R . Then (c i ) = (a) (bi ) for som e (bi) E Q . By the definition of R, there exist i 1 , i 2 ,. . . , in E 1 such tha t a i , ci E T for all i E 1 —  , i n } . Since T T is T -pure in T S , we have c i E a i S flT = a i T for all i E 1 — { i 1 , i 2 , . . . , i 7Z } . Hence w e can write c i = a i d i with some d i E T for each i E 1 — , i n } . Now define (x i ) by x i = b i for i = i 1 , i 2 ,... , i n and x i = d i for eac h i E I— {i 1 , i 2 , . . . , i n } . Then (x i ) E R, and hence (c i ) = (a)(x) E (a)R . This proves that R R is R -pure in R Q . Hence R is left pinjective b y Theorem 1 . Conversely, assume that R is left p-injective. Take an arbitrary a E T and set a i = a for all i . Then, by Theorem 1, we have (a)QflR = (a i )R . Now, let as E aS n T where s E S, and set s i = s for all i . The n (a)(s) E (a)Q n R =(a)R . Hence there exists (c i ) E R such tha t (ai)(si) = (a)(c) . By the definition of R, ci E T for almost all i . So , let a N E T . Then as = aa N E aT . This proves that aS n T = aT for al l a E T . Therefore T T is T -pure in T S . The proof of 2) <  > 3) is quite similar to that of 1) <  > 3), and so w e omit it . n The following example shows that a semiprime right and left pinjectiv e ring satisfying a polynomial identity need not be v on Neumann regular . Example 1 . Let K be a field and let T be the subring of the n x n ON NONSINGULAR p-INJECTIVE RINGS  45 9 full matrix ring M,(K) over K consisting of all matrices of the forr o a l a 2 . .  . .  an i a 1 a 2 with a 1 , a 2 , . . . , a n E K . Then T T is T -pure in TMn(K) . In fact, le t A be the matrix in (0 .1) and assume that a 1 = a 2 - - • • • ~ a, n _ 1 = 0 fc 1  .  c n and  O. Suppose that AB =  E T far some B E C 1 Mn(K) . Then we can writ e A = l0  a1 0 a,  ( CM . . . c n m  C m (d0 ,  dm ) in M n ._ rra+1 (K) with d m , . . . , d n E K . Therefore if we se t o\ X=  d n d m I then X E T and AB = AX . Thus T T is T -pure in T M n (K ) . Similarly w e can prove that T T is T -pure in M n (K) T . By Proposition 2, (M(K)jT) N and (M(K)IT)(N) are semiprime right and left p -injective rings satisfying a polynomial identity . However, we can easily see that these are no t von Neumann regular forn > 2 . The following example shows that a semiprime left p-injective rin g satisfying a polynomial identity need not be right p -injective . Example 2 . Let K be a field and consider the subrin g f a 0 0 T={ b a o  a,b,cEK } c 0 a 460  Y . HIRAN a 0 o 0  0 0 0 of .ZV1 3 (K) . Let A = 1 0 0 and B = 0 1 0 . Then it i s 1 0 0  0 0 0 easy to see that BA E T M 3 (K)A, but BA TA . Hence T T is no t T-pure in M3(K)T . Next, suppose that AX = B E T with A E T and X E M 3 (K) . If det(A) 0, then X = A' B E T . So, assume tha t det(A) = 0 and let x be the ( 1,1)- component of X . Then we can easily se e that B = A(xE) E AT, where E denotes the identity matrix in M 3 (K) . Thus, AM 3 (K) ~ T = AT . This implies T T is T -pure in T M 3 (K) . B y Proposition 2, semiprime PI-rings (M 3 (K)JT) N and (M3(K)T)(N) ar e left p-injective, but not right p -injective . The following shows that there are semiprime 7r - regular PI-rings whic h are neither right nor left p -injective . Example 3 . Let K be a field and let T be the algebra of uppe r triangular n x n matrices over K, where n ~ 1 . Then T T is not T -pur e in T Mn (K) . In fact, if {e} denotes the set of matrix units of Mn (K) , then e1raMn (K) n T = Ke11 +• • • + Ke~n ~ Ke1n = e 1n T . Hence , by Proposition 2, (M(K)IT) N and (M(K)fT)(N) are neither right pinjective nor left p -injective . However we can easily see that these ar e semiprime 7r - regular rings . Acknowledgement . 1 would like to thank the referee for his helpfu l comments and suggestions . Reference s 1. ARMENDARIZ E : P ., On semiprime P .I . - algebras over commutativ e regular rings, Pacific J . Math . 66 (1976), 23-28 . 2. DISCHINGER F ., Sur les anneaux fortement 7r-réguliers, C . R . Acad . Sci . Paris 283 (1976), 571-573 . 3. FISHER J . W . AND SNIDER R . L ., On the von Neumann regularit y of rings with regular prime factor rings, Pacific c J . Math . 54 (1974) , 135-144 . 4. GGGDEARL K . R ., "Ring Theory," Marcel Dekker, Inc ., New Yor k and Basel, 1976 . 5. HANNAH J ., Quotient rings of semiprirne rings with bounded index , Glasgow Math . J . 23 (1982), 53-64 . 6. HERSTEIN I . N ., "Tapies in Ring Theory," The University of Chi - cago Press, 1969 . ON NONSINGULAR p-INJECTIVE RINGS  46 1 7. HIRANO Y ., Some characterizations of 7r-regular rings of bounde d index, Math . J . 01cayama Univ . 32 (1990), 97-101 . 8. IKEDA M. AND NAKAYAMA T ., on some characteristic properties o f quasi-Frobenius and regular rings, Proc . Amer . Math . Soc. 5 (1954) , 15-18 . 9. JACOBSON N ., " Structure of Rings,," Amer . Math . Soc . Colloq . Publ . 37, Providence, R. I ., 1964 . 10. KAPLANSKY I ., Topological representation of algebras, II, Trans . Amer . Math . Soc . 68 (1950), 62-75 . 11. YUE CHI MING R ., on (von Neumann) regular rings, Proc . Edin - burgh Math. Soc . 19 (1974), 89-91 . Department of Mathematic s Okayama Universit y Okayama 70 0 JAPA N Primera versid rebuda el 5 de Setembre de 1994 , darrera versid rebuda el 14 d'Octubre de 1994