2. Since A[b] is a subring of B, it is an integral domain. Thus if bz = 0 and b = 0, then z = 0. 3. Any linear transformation on a finite-dimensional vector space is injective iff it is surjective. Thus if b ∈ B and b = 0, there is an element c ∈ A[b] ⊆ B such that bc = 1. Therefore B is a field. 4. Since P is the preimage of Q under the inclusion map of A into B, P is a prime ideal. The map a ...

The arithmetic behavior of the partition function has been of great interest. For example, we have the famous Ramanujan congruences p(5n+ 4) ≡ 0 (mod 5), p(7n+ 5) ≡ 0 (mod 7), p(11n+ 6) ≡ 0 (mod 11) for every n ≥ 0. In a celebrated paper Ono [13] treated this type of congruence systematically. Combining Shimura’s theory of modular forms of half-integral weight with results of Serre on modular f...

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, ${0})$ that are closed under the natural metric, but has no prime ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Togethe...

We establish upper and lower bounds for the number of solutions to A+B = C in relatively prime integers A,B,C with max(|A|, |B|, |C|) ≤ N and Rad(A) ≤ N,Rad(B) ≤ N,Rad(C) ≤ N, valid when 0 < a, b, c ≤ 1. The lower bound is Ω(Na+b+c−1(logN)−2), and the upper bound is of the form O(Na+b+c−1+ + N ), for any fixed > 0. In particular, these bounds match up to N factors so long as a+ b+ c ≥ 2, verify...

Abstract There may be seven hadronic molecular states. We construct their corresponding interpolating currents and calculate masses decay constants using QCD sum rules. Based on these results, we relative production rates in \Lambda_b^0 decays current algebra, that is, {\cal{B}}(\Lambda_b^0 \to P_c K^-):{\cal{B}}(\Lambda_b^0 P_c^\prime K^-) , where are ...