New criteria for a ring to have a semisimple left quotient ring V. V. Bavula (University of Sheffield) ∗ ∗ V. V. Bavula, New criteria for a ring to have a semisim- ple left quotient ring. Arxiv:math.RA:1303.0859. talk-NewCrit-SS-lQuot.tex 1
Goldie’s Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring . The aim of the talk is to give four new criteria (using a completely different approach and new ideas). The First Criterion is based on the recent fact that for an arbitrary ring R the set M of maximal left denominator sets of R is a non- empty set . The Second Criterion is given via the mini- mal primes of R and goes further then the First one in the sense that it describes explicitly the maximal left denominator sets S via the mini- mal primes of R . 2
The Third Criterion is close to Goldie’s Cri- terion but it is easier to check in applications (basically, it reduces Goldie’s Theorem to the prime case). The Fourth Criterion is given via certain left denominator sets. 3
R is a ring with 1, R ∗ is its group of units, C = C R is the set of regular elements of R , Q = Q l,cl ( R ) := C − 1 R is the left quotient ring (the classical left ring of fractions ) of R (if it exists), Ore l ( R ) is the set of left Ore sets S (i.e. for all s ∈ S and r ∈ R : Sr ∩ Rs ̸ = ∅ ), Den l ( R, a ) is the set of left denominator sets S of R with ass( S ) = a where a is an ideal of R and ass( S ) := { r ∈ R | sr = 0 for some s ∈ S } (i.e. S ∈ Ore l ( R ), and rs = 0 implies s ′ r = 0 for some s ′ ∈ S ), max . Den l ( R ) is the set of maximal left de- nominator sets of R (it is always a non-empty set). 4
A ring R is a left Goldie ring if (i) R satisfies ACC for left annihilators, (ii) R contains no infinite direct sums of left ideals. Thm (Goldie, 1958, 1960) . A ring R has a semisimple left quotient ring iff R is a semiprime left Goldie ring. Lessieur and Croisot (1959): prime case. Question : When Q does exist and is a left Artinian ring? Answer: Small (1966), Robson (1967), Tachikawa (1971), Hajarnavis (1972) and Bavula (2012). In all the proofs of the criteria above Goldie’s Thm is used. 5
First Criterion Recently, I introduced several new concepts: the largest left quotient ring of a ring, the largest regular left Ore set of a ring, a max- imal left denominator set of a ring, the left localization radical of a ring, a left localization maximal ring, the core of an Ore set . Their universal nature naturally leads to the present criteria for a ring to have a semisimple left quo- tient ring. The First Criterion is given via the set M := max . Den l ( R ). Theorem (B. 2013, First Criterion) A ring R has a semisimple left quotient ring Q iff (a) M is a finite set, (b) ∩ S ∈M ass( S ) = 0 , (c) for each S ∈ M , the ring S − 1 R is a simple left Artinian ring. S ∈M S − 1 R . In this case, Q ≃ ∏ 6
Second Criterion The Second Criterion is given via the minimal primes of R and certain explicit multiplicative sets associated with them. On the one hand, the Second Criterion stands between Goldie’s Theorem and the First Cri- terion in terms how it is formulated. On the other hand, it goes further then the First Criterion and Goldie’s Theorem in the sense that it describes explicitly the maximal left denominator sets and the left quotient ring of a ring with a semisimple left quotient ring. 7
Theorem (B. 2013, Second Criterion) Let R be a ring. The following statements are equivalent. 1. The ring R has a semisimple left quotient ring Q . 2.(a) R is a semiprime ring. (b) The set Min( R ) of minimal primes of R is a finite set. (c) For each p ∈ Min( R ) , the set S p := { c ∈ R | c + p ∈ C R/ p } is a left denominator set of the ring R with ass( S p ) = p . (d) For each p ∈ Min( R ) , the ring S − 1 R is a p simple left Artinian ring. If one of the two equivalent conditions holds then max . Den l ( R ) = { S p | p ∈ Min( R ) } and Q ≃ p ∈ Min( R ) S − 1 R . ∏ p 8
Third Criterion The Third Criterion can be seen as a ‘weak’ version of Goldie’s Theorem in the sense that the conditions are ‘weaker’ than that of Goldie’s Theorem. In applications, it is ‘easier’ to verify whether a ring satisfies the conditions comparing with Goldie’s Theorem as the Third Criterion ‘re- duces’ Goldie’s Theorem essentially to the prime case and reveals the ‘local’ nature of Goldie’s Theorem. 9
Theorem (B. 2013, Third Criterion) Let R be a ring. The following statements are equivalent. 1. The ring R has a semisimple left quotient ring Q . 2. The ring R is a semiprime ring with | Min( R ) | < ∞ and, for each p ∈ Min( R ) , the ring R/ p is a left Goldie ring. The condition | Min( R ) | < ∞ can be replaced by any of the four equivalent conditions of the next Theorem , e.g. ‘ the ring R has a.c.c. on annihilator ideals .’ 10
Theorem The following conditions on a semiprime ring R are equivalent. 1. R R R has finite uniform dimension. 2. | Min( R ) | < ∞ . 3. R has finitely many annihilator ideals. 4. R has a.c.c. on annihilator ideals. 11
Fourth Criterion Theorem (B. 2013, Fourth Criterion) Let R be a ring. The following statements are equivalent. 1. The ring R has a semisimple left quotient ring Q . 2. There are left denominator sets S ′ 1 , . . . , S ′ n of R such that the rings R i := S − 1 R , i = 1 , . . . , n , i are simple left Artinian rings and the map n n R i , r �→ ( r 1 , . . . , r ∏ ∏ σ := σ i : R → 1) , i =1 i =1 is an injection where σ i : R → R i , r �→ r 1 . 12
The maximal left quotient rings of a finite direct product of rings . Theorem (B. 2013) Let R = ∏ n i =1 R i be a direct product of rings R i . Then for each i = 1 , . . . , n , the map max . Den l ( R i ) → max . Den l ( R ), S i �→ R 1 × · · · × S i × · · · × R n , is an injection. Moreover, max . Den l ( R ) = ⨿ n i =1 max . Den l ( R i ) in the above sense, i.e. max . Den l ( R ) = { S i | S i ∈ max . Den l ( R i ) , i = 1 , . . . , n } , S − 1 R ≃ S − 1 R i , i i ass R ( S i ) = R 1 × · · · × ass R i ( S i ) × · · · × R n . 13
A bijection between max . Den l ( R ) and max . Den l ( Q l ( R )). Proposition (B. 2013) . Let R be a ring, S l be the largest regular left Ore set of the ring R , Q l := S − 1 R be the largest left quotient ring of l the ring R , and C be the set of regular elements of the ring R . Then 1. S l ⊆ S for all S ∈ max . Den l ( R ) . In particu- lar, C ⊆ S for all S ∈ max . Den l ( R ) provided C is a left Ore set. 2. Either max . Den l ( R ) = {C} or, otherwise, C ̸∈ max . Den l ( R ) . 3. The map max . Den l ( R ) → max . Den l ( Q l ) , S �→ SQ ∗ l = { c − 1 s | c ∈ S l , s ∈ S } , 14
�→ σ − 1 ( T ) is a bijection with the inverse T r 1 , and SQ ∗ where σ : R → Q l , r �→ is the l sub-semigroup of ( Q l , · ) generated by the set S and the group Q ∗ l of units of the ring Q l , and S − 1 R = ( SQ ∗ l ) − 1 Q l . 4. If C is a left Ore set then the map max . Den l ( R ) → max . Den l ( Q ) , S �→ SQ ∗ = { c − 1 s | c ∈ C , s ∈ S } , �→ σ − 1 ( T ) is a bijection with the inverse T 1 , and SQ ∗ is the r where σ : R → Q , r �→ sub-semigroup of ( Q, · ) generated by the set S and the group Q ∗ of units of the ring Q , and S − 1 R = ( SQ ∗ ) − 1 Q . 15
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