Does the nfc ring work with this rfid reader?
According to the NFC Ring FAQ page, http://store.nfcring.com/pages/faqs, they use the NXP NTAG203 chip as NFC tag in the ring. According to NXP the NTAG203 operates in the 13.56 MHz frequency band, http://www.nxp.com/products/identification_and_security/smart_label_and_tag_ics/ntag/series/NTAG203.html.

The RFID reader you have linked to operates in the 125 kHz frequency band, https://www.sparkfun.com/products/11828.

This means that the ring and reader is not compatible with each other. You need a reader that operates in the 13.56 MHz frequency band to be able to read the tag in the ring.