A homomorphism is one to one if and only if its kernel consists of just the identity. Every homomorphism is one to one. A homomorphism may have an empty kernel. Every factor group of a cyclic group is cyclic. A divisor of 0 in a commutative ring with unity can have no multiplicative inverse.
Every field is also a ring.
The nonzero elements of a field for a group under multiplication. A non-commutative ring. A commutative ring without a unit. An infinite ring with characteristic 2. A finite field.