Skew NN-Derivations on Semiprime Rings

For a ring $R$ with an automorphism $\sigma$, an $n$-additive mapping $\Delta:R\times R\times... \times R \rightarrow R$ is called a skew $n$-derivation with respect to $\sigma$ if it is always a $\sigma$-derivation of $R$ for each argument. Namely, it is always a $\sigma$-derivation of $R$ for the argument being left once $n-1$ arguments are fixed by $n-1$ elements in $R$. In this short note, starting from Bre\v{s}ar Theorems, we prove that a skew $n$-derivation ($n\geq 3$) on a semiprime ring $R$ must map into the center of $R$.Comment: 8 page

Hunting for Extremely Faint Planetary Nebulae in the SDSS Spectroscopic Database

Using ~1,700,000 target- and sky-fiber spectra from the SDSS, we have carried out a systematic search for Galactic planetary nebulae (PNe) via detections of the [OIII] 4959, 5007 lines. Thanks to the excellent sensitivity of the SDSS spectroscopic surveys, this is by far the deepest search for PNe ever taken, reaching a surface brightness of the [OIII] 5007 line down to about 29.0 mag./arcsec^2. The search recovers 13 previously known PNe in the Galactic Caps. In total, 44 new PN candidates are identified, including 7 candidates of multiple detections and 37 candidates of single detection. The 7 candidates of multiple detections are all extremely large (between 21' - 154') and faint, located mostly in the low Galactic latitude region and with a kinematics similar to disk stars. After checking their images in Ha and other bands, three of them are probably HII regions, one is probably associated with a new supernova remnant, another one is possibly a true PN, and the remaining two could be either PNe or supernova remnants. Based on sky positions and kinematics, 7 candidates of single detection probably belong to the halo population. If confirmed, they will increase the number of known PNe in the Galactic halo significantly. All the newly identified PN candidates are very faint, with a surface brightness of the [OIII] 5007 line between 27.0 - 30.0 mag./arcsec^2, very challenging to be discovered with previous techniques and thus may greatly increase the number of "missing" faint PNe. Our results demonstrate the power of large scale fiber spectroscopy in hunting for ultra-faint PNe and other types of emission line nebulae. Combining the large spectral databases provided by the SDSS and other on-going projects (e.g. the LAMOST Galactic surveys), it is possible to build a statistically meaningful sample of ultra-faint, large, evolved PNe, thus improving the census of Galactic PNe.Comment: 23 pages, 1 table and 16 figures. MNRAS accepted. High resolution version and online-only material may be found at http://kiaa.pku.edu.cn/DSSGAC/YL13_pne_sdss.pd