I got a surgical stainless ring from Amazon for $15 and haven't had any issues. Actually bought a second one once I was sure of the fit in case I ever lose the first.
My wife got her engagement and wedding bands at Sam's club for under $800 total. Engagement is diamond and has had no issues, but the wedding bands are some fake stones that scuffed in less than a year and lost their luster. She didn't research at all, just bought what she liked. So if you are going with stones, don't just be cheap - frugal would be a quality "fake" or lower quality/flawed diamond. A lot recommend moissanite, but I can't speak to it myself.