Prove or disprove: If E is a subring of an ordered integral domain D, and E is also an integral domain, then E is ordered.

Trying to satisfy the ordered integral domain definiton:

since E is a ring ...it is closed under addition..then it's positive elements certainly are.

also E's nonzero elements are closed under multiplication because it is an integral domain.

help ...i don't know how to complete and write a proper proof for this.