I'm having a difficult time in my Abstract Algebra course with the concept of left/right identity and inverse elements for a binary operation. On a recent assignment (which I will be altering slightly to maintain academic integrity), my professor defined several binary operations and asked us to define whether each is commutative, associative, which elements have right identities, which have left identities, and which elements have left/right inverses. We do not have to justify our answers but should list the identity/inverse elements.
I have a rudimentary understanding of identity/inverse elements to the extent of:
Right identity element = e such that a Δ e = a;
Left identity element = e such that e Δ a = a;
Left/Right inverse element = x such that x Δ a = a Δ x = e
1.) Consider the binary operation defined as a Δ b = a + 3b where a, b ∈ R.
My attempt at a solution: It is not commutative and it is not associative. Right identity: a Δ e = a + 3e, so e = 0 Left identity: e Δ a = e + 3a, so e = -2a (??)
2.) Now, consider x Δ y = x + y + xy for x, y ∈ R.
My attempt: It is commutative and it is associative. Right identity: e = 0 Left identity: e = 0
0 has no multiplicative inverse, but it is its own additive inverse. Would my right and left inverse elements then both be 0?
After this, I could try and find the left inverse element using the formula I noted above, though I feel like I'm misunderstanding some fundamentals of this problem. My professor mentioned something about a "true identity element," which is when both the left and right identity elements are the same, and that when they are not the same then there can only be one (?). Unfortunately, my notes are failing to provide help with this problem. Any insight is appreciated--thanks!