basic facts about ordered rings
Throughout this entry, is an ordered ring.
If with , then .
The contrapositive will be proven.
Let with . Note that . Thus,
If and has a characteristic, then it must be .
Suppose not. Let be a positive integer such that . Since , it must be the case that .
Let with . By the previous lemma, , a contradiction. ∎
If with and with , then .
Note that and . Since , . Thus,
Suppose further that is a ring with multiplicative identity . Then .
Suppose that . Since is an ordered ring, it must be the case that . By the previous lemma, . Thus, , a contradiction. ∎
|Title||basic facts about ordered rings|
|Date of creation||2013-03-22 16:17:21|
|Last modified on||2013-03-22 16:17:21|
|Last modified by||Wkbj79 (1863)|