(a) A surety who has paid his principal's note, bill, bond or other written obligation, may either sue his principal for reimbursement or sue his principal on the instrument and may maintain any action or avail himself of any remedy which the creditor himself might have had against the principal debtor. No assignment of the obligation to the surety or to a third-party trustee for the surety's benefit shall be required.
(b) The word "surety" as used herein includes a guarantor, accommodation maker, accommodation indorser, or other person who undertakes liability for the written obligation of another.