Under Arizona law a landlord is not required to accept a short or partial rent payment. However, per A.R.S. &33-1371 "if the landlord does accept the lesser amount (than full rent), the landlord loses their right to seek eviction of the resident". Keep in mind that this does include scheduled semi-monthly payment plans, where residents are regularly permitted to pay the rent on the 1st and the 15th without penalty. Acceptance of half the rent at the beginning of the month is an automatic bar to evicting the resident even if they do not make the second payment on the 15th.
The graver problem, however is when landlords hold onto payments from tenants to see if they can come up with the remainder of the funds. Unfortunately even if you do not deposit the payment from the tenant, the Courts can and will rule that you have accepted it. Acceptance can occur even without a specific action, because holding onto the money deprives the resident from its use. When the resident is deprived of its use by the landlord, then the landlord functionally has accepted it.
You can preserve your rights to enforce the lease (and evict) if you enter into a partial-payment (or promise-to-pay) non-waiver agreement. This specialized legal notice lists three crucial elements to preserve a landlord's rights: a) the short payment is identified and credited to the account; b) the remaining balance is scheduled for payment; and (c) a declaration is made that the landlord is not waiving the right to terminate the resident for paying less than the full amount due. Good agreements also include the language of the non-payment Five Day Notice (A.R.S. & 33-1368(B)) in order to avoid issuing a new notice if the scheduled payments are missed.
When it comes to rent collections it is often best to have a Phoenix Property Manager such as RPM West Valley Phoenix being the ones collecting. With management services you don't have to worry about being "too nice" and giving the tenant extra time to pay often times forgetting that a tenant who pays late this month most likely won't have the funds to pay on time next month and the cycle will continue.