Characterizing Jordan derivations of matrix rings through zero products

Let \Mn be the ring of all $n \times n$ matrices over a unital ring $\mathcal{R}$, let $\mathcal{M}$ be a 2-torsion free unital \Mn-bimodule and let D:\Mn\rightarrow \mathcal{M} be an additive map. We prove that if D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)=0 whenever \A,\B\in \Mn are such that \A\B=\B\A=0, then D(\A)=\delta(\A)+\A D(\textbf{1}), where \delta:\Mn\rightarrow \mathcal{M} is a derivation and $D(\textbf{1})$ lies in the centre of $\mathcal{M}$. It is also shown that $D$ is a generalized derivation if and only if D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)-\A D(\textbf{1})\B-\B D(\textbf{1})\A=0 whenever \A\B=\B\A=0. We apply this results to provide that any (generalized) Jordan derivation from \Mn into a 2-torsion free \Mn-bimodule (not necessarily unital) is a (generalized) derivation. Also, we show that if \varphi:\Mn\rightarrow \Mn is an additive map satisfying \varphi(\A \B+\B \A)=\A\varphi(\B)+\varphi(\B)\A \quad (\A,\B \in \Mn), then \varphi(\A)=\A\varphi(\textbf{1}) for all \A\in \Mn, where $\varphi(\textbf{1})$ lies in the centre of \Mn. By applying this result we obtain that every Jordan derivation of the trivial extension of \Mn by \Mn is a derivation.Comment: To appear in Mathematica Slovac

Kajian Penggunaan Pupuk Organik Sampah Kota Makassar Pada Tanaman Cabai (Capsicum Annum L)

An assessment aimed to find out the benefit application at organic fertilizer from city garbage on red pepperplanting in lowland-after rice with inceptisols Bajeng-Gowa district, South Sulawesi. The study was carried out fromJune to October 2000. Assessment was set in randomized block design with nine treatments and three replications.Treatments consist to several level of organic fertilizers from city garbage and combination of inorganic and organicfertilizers. Result showed that application of landfill\u27s organic fertilizer (LOF) and its combination with inorganicfertilizer were useful positively in term of growth and yield improvement of red pepper, as well as increased inincome. Application of 50 kg urea + 100 kg SP-36 + 100 kg KCl + 6,0 t LOF/ha resulted the highest production of redpepper (11,872 kg/ha) with net income of Rp. 33.132.000 and VCR of 3,0. The higher rate application of landfill\u27sorganic fertilizer, the more benefit would be gained. Application of 10,0 t LOF gave fresh fruit production of 9,616kg/ha, higher than that recommended fertilizer of 150 kg urea + 150 kg SP-36 + 150 kg KCl / ha (8,706 kg/ha), andyielded net income of Rp. 23.990.000, and VCR of 1,8. Subsequently, application of 50 kg urea + 20 t LOF / ha stillindicated good yield, fresh fruit production reached was 7,618 kg/ha, with net income of Rp. 22.443.000 / ha, andVCR of 2,5. Recommended fertilizer on red pepper planting in low land after rice with Inceptisols Soil in Bajeng was50 kg Urea + 2-6 t OF TPA/ha.Key words : garbage, organic fertilizers, wetland, Capsicum annum L., South Sulawesi Suatu kajian yang bertujuan untuk melihat manfaat penggunaan pupuk organik sampah dari tempatpembuangan akhir (TPA) pada tanaman cabai telah dilakukan di lahan sawah sesudah padi, pada tanah InceptisolBajeng, Gowa, Sulawesi Selatan. Kajian berlangsung bulan Juni sampai dengan Oktober 2000. Kajian disusunmenurut Rancangan Acak Kelompok dengan sembilan perlakuan dan tiga ulangan (petani representatif dari ulangan).Perlakuan terdiri dari berbagai takaran pupuk organik sampah TPA dengan kombinasi pupuk anorganik dan organik.Hasil kajian menunjukkan bahwa aplikasi pupuk organik (PO) TPA dan kombinasinya dengan pupuk anorganikpositif terhadap perbaikan pertumbuhan dan hasil cabai, serta pendapatan. Penggunaan 50 kg urea + 100 kg SP-36 +100 kg KCl + 6,0 t PO TPA tunggal menghasilkan produksi cabai tertinggi (11.872 kg/ha) dengan keuntungan sebesarRp. 33.132.000 dan VCR 3,0. Aplikasi 10 t PO TPA/ha menghasilkan produksi buah segar 9.616 kg/ha, lebih tinggidaripada hasil yang diperoleh dengan penggunaan paket pupuk rekomendasi, 150 kg urea + 150 kg SP-36 + 150 kgKCl/ha (8.706 kg/ha), dengan tingkat keuntungan Rp. 23.990.000, dan nilai VCR sebesar 1,8. Selanjutnya, aplikasi 50kg urea + 2,0 t PO TPA/ha mampu memberi hasil yang cukup menggembirakan, produksi buah segar sebanyak 7.618kg/ha, keuntungan sebesar Rp. 22.443.000 /ha, dan nilai VCR 2,5. Rekomendasi pemupukan pada cabai yangdiharapkan dapat bermanfaat sebesar-besarnya bagi petani adalah 50 kg Urea + 2-6 t PO TPA/ha

Abelian Vectors and Self-Dual Tensors in Six-Dimensional Supergravity

In this note we describe the most general coupling of {\it abelian} vector and tensor multiplets to six-dimensional $(1,0)$ supergravity. As was recently pointed out, it is of interest to consider more general Chern-Simons couplings to abelian vectors of the type $H^{r}=d B^{r}-1/2 c^{rab} A^{a}d A^{b}$, with $c^{r}$ matrices that may not be simultaneously diagonalized. We show that these couplings can be related to Green-Schwarz terms of the form $B^r c_r^{ab} F^a F^b$, and how the complete local Lagrangian, that embodies factorized gauge and supersymmetry anomalies (to be disposed of by fermion loops) is uniquely determined by Wess-Zumino consistency conditions, aside from an arbitrary quartic coupling for the gauginos.Comment: LaTeX file, 10 pages. Some equations, previously written with wrong notations, are correcte