bath这个要加定冠词吗?新概念一册81课Carol回答时说：He's having a bath.这个‘a’应该不用加

bath这个要加定冠词吗?
新概念一册81课
Carol回答时说：He's having a bath.
这个‘a’应该不用加的吧
另外第83课里
原句是：
I've already had lunch.
I had lunch at half past twelve.
将它改成:I've already had lunch at half past twelve。
这个不影响它的完成时结构吧
还有I've already had lunch的形容词already是现在完成时必须要加的吗？
还是不加也可以？