As I understand it, to be a Republic: decision makers are elected and the rule of law via is codified via a constitution. If there is but a single party, and officials are elected by the people, is it not possible that this government be a Republic? or does a single party somehow preclude being a Republic?
There are two ways that you can have a "one-party" government:
1) There is one party (such as the Communist Party in most countries that it ruled) that has a centralized system for nominating just one candidate for each elected position. These countries called themselves "republics", but it was obvious that the "people" had no choice about whom they could elect.
2) There is one party, which does not have any organized opposition, but multiple candidates are still encouraged to run for office. The candidates are chosen via primaries or other bottom-up selection processes. (For example, the United States after the collapse of the Federalist Party, and before the Democratic-Republican party split into Jacksonian and Whig factions.) These countries are true republics. These countries are not obviously different from republics that have no parties.
What is a Republic?
Political terms like 'republic' are not well defined. They are philosophic or political, not precise technical terms. Generally, a republic has elections to determine leaders. Beyond that, you will encounter a lot of debate about what a republic is or isn't.
There may or may not be well-institutionalized rule of law (the republics of ancient Greece didn't). There may or may not be elected officials (again, ancient Greek republics were direct democracies). Some people will insist on less tangible qualities (like fair or free elections, or citizen involvement).
To answer your question directly - it should be obvious from your definition that being a republic has nothing to do with being a single-party system. If that seems counter-intuitive to you, you might refine your definition of "republic" or your intuition about a single party systems.