Make sure your partner is on the same page. If you're both male, chances are you'll be okay.. if you discuss it with him first. And don't say that you want to go cheap for the sake of going cheap - if you cheap on the engagement ring, you usually need to make up for it on the wedding ring. Every partner deserves the best regardless of their gender identification.
However, if your partner is female.. the lady wants to show off her engagement ring; this is her special time to shine. The only thing that should be more dazzling is the wedding ring; this, she will be showing people for the rest of her life.
Go cheap at your own peril.
Though it won't help you much in this situation, one thing I did was to get a cubic zirconium necklace. It was cheap, but it was really pretty and different, so she loved it. Our engagement was short, though, and she now sports an Italian-made ring (Gelin Abaci) with a huge sapphire, and diamonds. She still gets comments on it, and loves it.