Activity 9.7. Let R be a ring, and let S be a subring of R. Which of the following conjectures do
you think are true, and which do you think are false? Whenever possible, provide brief arguments
or examples to justify your answers.
(a) If R has a multiplicative identity, then S has a multiplicative identity.
(b) If S has a multiplicative identity, then R has a multiplicative identity.
(c) If both R and S have a multiplicative identity, then $1_R = 1_S$.
(d) If R is commutative, then S is commutative.
(e) If S is commutative, then R is commutative.
(f) If R and S both have identity and x is a unit in R, then x is a unit in S. (Does your answer
depend on whether $1_R = 1_S$ ?)
(g) If R and S both have identity and x is a unit in S, then x is a unit in R. (Does your answer
depend on whether $1_R = 1_S$ ?)
(h) If R is a field, then S is a field.
(i) If S is a field, then R is a field.
(j) If R is an integral domain, then S is an integral domain.
(k) If S is an integral domain, then R is an integral domain.
Activities 06 and 07 demonstrate that the relationshi
