Rings Over Which Cyclics are Direct Sums of Projective and CS or Noetherian

R is called a right WV -ring if each simple right R-module is injective relative to proper cyclics. If R is a right WV -ring, then R is right uniform or a right V -ring. It is shown that for a right WV-ring R, R is right noetherian if and only if each right cyclic module is a direct sum of a projective module and a CS or noetherian module. For a finitely generated module M with projective socle over a V -ring R such that every subfactor of M is a direct sum of a projective module and a CS or noetherian module, we show M = X \oplus T, where X is semisimple and T is noetherian with zero socle. In the case that M = R, we get R = S \oplus T, where S is a semisimple artinian ring, and T is a direct sum of right noetherian simple rings with zero socle. In addition, if R is a von Neumann regular ring, then it is semisimple artinian.Comment: A Para\^itre Glasgow Mathematical Journa

The Structure of a Low-Metallicity Giant Molecular Cloud Complex

To understand the impact of low metallicities on giant molecular cloud (GMC) structure, we compare far infrared dust emission, CO emission, and dynamics in the star-forming complex N83 in the Wing of the Small Magellanic Cloud. Dust emission (measured by Spitzer as part of the S3MC and SAGE-SMC surveys) probes the total gas column independent of molecular line emission and traces shielding from photodissociating radiation. We calibrate a method to estimate the dust column using only the high-resolution Spitzer data and verify that dust traces the ISM in the HI-dominated region around N83. This allows us to resolve the relative structures of H2, dust, and CO within a giant molecular cloud complex, one of the first times such a measurement has been made in a low-metallicity galaxy. Our results support the hypothesis that CO is photodissociated while H2 self-shields in the outer parts of low-metallicity GMCs, so that dust/self shielding is the primary factor determining the distribution of CO emission. Four pieces of evidence support this view. First, the CO-to-H2 conversion factor averaged over the whole cloud is very high 4-11 \times 10^21 cm^-2/(K km/s), or 20-55 times the Galactic value. Second, the CO-to-H2 conversion factor varies across the complex, with its lowest (most nearly Galactic) values near the CO peaks. Third, bright CO emission is largely confined to regions of relatively high line-of-sight extinction, A_V >~ 2 mag, in agreement with PDR models and Galactic observations. Fourth, a simple model in which CO emerges from a smaller sphere nested inside a larger cloud can roughly relate the H2 masses measured from CO kinematics and dust.Comment: 17 pages, 10 figures (including appendix), accepted for publication in the Astrophysical Journa

Unusually Luminous Giant Molecular Clouds in the Outer Disk of M33

We use high spatial resolution (~7pc) CARMA observations to derive detailed properties for 8 giant molecular clouds (GMCs) at a galactocentric radius corresponding to approximately two CO scale lengths, or ~0.5 optical radii (r25), in the Local Group spiral galaxy M33. At this radius, molecular gas fraction, dust-to-gas ratio and metallicity are much lower than in the inner part of M33 or in a typical spiral galaxy. This allows us to probe the impact of environment on GMC properties by comparing our measurements to previous data from the inner disk of M33, the Milky Way and other nearby galaxies. The outer disk clouds roughly fall on the size-linewidth relation defined by extragalactic GMCs, but are slightly displaced from the luminosity-virial mass relation in the sense of having high CO luminosity compared to the inferred virial mass. This implies a different CO-to-H2 conversion factor, which is on average a factor of two lower than the inner disk and the extragalactic average. We attribute this to significantly higher measured brightness temperatures of the outer disk clouds compared to the ancillary sample of GMCs, which is likely an effect of enhanced radiation levels due to massive star formation in the vicinity of our target field. Apart from brightness temperature, the properties we determine for the outer disk GMCs in M33 do not differ significantly from those of our comparison sample. In particular, the combined sample of inner and outer disk M33 clouds covers roughly the same range in size, linewidth, virial mass and CO luminosity than the sample of Milky Way GMCs. When compared to the inner disk clouds in M33, however, we find even the brightest outer disk clouds to be smaller than most of their inner disk counterparts. This may be due to incomplete sampling or a potentially steeper cloud mass function at larger radii.Comment: Accepted for Publication in ApJ; 7 pages, 4 figure

Arm & Interarm Star Formation in Spiral Galaxies

We investigate the relationship between spiral arms and star formation in the grand-design spirals NGC 5194 and NGC 628 and in the flocculent spiral NGC 6946. Filtered maps of near-IR (3.6 micron) emission allow us to identify "arm regions" that should correspond to regions of stellar mass density enhancements. The two grand-design spirals show a clear two-armed structure, while NGC 6946 is more complex. We examine these arm and interarm regions, looking at maps that trace recent star formation - far-ultraviolet (GALEX NGS) and 24 micron emission (Spitzer, SINGS) - and cold gas - CO (Heracles) and HI (Things). We find the star formation tracers and CO more concentrated in the spiral arms than the stellar 3.6 micron flux. If we define the spiral arms as the 25% highest pixels in the filtered 3.6 micron images, we find that the majority (60%) of star formation tracers occurs in the interarm regions; this result persists qualitatively even when considering the potential impact of finite data resolution and diffuse interarm 24 micron emission. Even with a generous definition of the arms (45% highest pixels), interarm regions still contribute at least 30% to the integrated star formation rate tracers. We look for evidence that spiral arms trigger star or cloud formation using the ratios of star formation rate (SFR, traced by a combination of FUV and 24 micron emission) to H_2 (traced by CO) and H_2 to HI. Any enhancement of SFR / M(H_2) in the arm region is very small (less than 10%) and the grand design spirals show no enhancement compared to the flocculent target. Arm regions do show a weak enhancement in H_2/HI compared to the interarm regions, but at a fixed gas surface density there is little clear enhancement in the H_2/HI ratio in the arm regions. Thus, it seems that spiral arms may only act to concentrate the gas to higher densities in the arms.Comment: 11 pages, 9 Figures, accepted by Ap