Absent a writing detailing the terms and conditions of the partial payment, it seems that Arizona's statue on this is clear:
33-1371 A. A landlord is not required to accept a partial payment of rent or other charges. A landlord accepting a partial payment of rent or other charges retains the right to proceed against a tenant only if the tenant agrees in a contemporaneous writing to the terms and conditions of the partial payment with regard to continuation of the tenancy. The written agreement shall contain a date on which the balance of the rent is due. The landlord may proceed as provided in article 4 of this chapter and in title 12, chapter 8 against a tenant in breach of this agreement or any other breach of the original rental agreement. If the landlord has provided the tenant with a notice of failure to pay rent as specified in section 33-1368, subsection B prior to the completion of the agreement for partial payment, no additional notice under section 33-1368, subsection B is required in case of a breach of the partial payment agreement.
B. Except as specified in subsection A of this section, acceptance of rent, or any portion thereof, with knowledge of a default by tenant or acceptance of performance by the tenant that varied from the terms of the rental agreement or rules or regulations subsequently adopted by the landlord constitutes a waiver of the right to terminate the rental agreement for that breach.
You may want to verify this with an Arizona landlord/tenant attorney. Failing to pay the entire amount of rent due and owing could constitute a breach of your original lease against the landlord. Moreover, if you suffered an injury as a result of your landlord's negligence, you may have other causes of action against the landlord.
DISCLAIMER: Brandy A. Peeples is licensed to practice law in the State of Maryland. This answer is being provided for informational purposes only and the laws of your jurisdiction may differ. This answer based on general legal principles and is not intended for the purpose of providing specific legal advice or opinions. Under no circumstances does this answer constitute the establishment of an attorney-client relationship. For legal advice relating to your specific situation, I strongly urge you to consult with an attorney in your area. NO COMMUNICATIONS WITH ME ARE TO BE CONSTRUED AS ARISING FROM AN ATTORNEY-CLIENT RELATIONSHIP AND NO ATTORNEY-CLIENT RELATIONSHIP WILL BE ESTABLISHED WITH ME UNLESS I HAVE EXPRESSLY AGREED TO UNDERTAKE YOUR REPRESENTATION, WHICH INCLUDES THE EXECUTION OF A WRITTEN AGREEMENT OF RETAINER.
A LL may evict if rent is not paid. If you paid LL cash, and received no written receipt or agreement--donuts to dollars the LL will say you paid nothing and you've nothing to show that you did pay. Additionally, depending on when you did pay, if you did not pay in full, and you stil owe rent, the landlord is privileged (in fact, many would argue that a good landlord must) begin the eviction process.
FYI oral contracts are binding in every state I am aware of--proving the provisions of an oral contract is very difficult, thats what makes them hard to pursue in court.
DO NOT rely on what "someone told you" or what you heard a landlord could or could not do regarging the validity of a contract unless that person was an AZ attorney, working on your behalf.
Recommend you contact an Apache Junction area landlord tenant attorney ASAP if you want to remain in that property.
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The statute cited here regarding acceptance of partial payments should help you, but only if your justice of the peace knows about it. Bring a copy with you to your forcible detainer hearing along with any physical evidence you have that your landlord accepted a partial payment. I assume you are talking about a partial payment for the August rent, because if he accepted a partial payment in July and you paid nothing in August, then he absolutely can evict you.