Brainstorm topics on algebra. For example, if you are interested in C*-algebras, a Banach algebra with involution, look up articles about the research on C*-algebras. Make a list of questions that you have about the topic. For example, is it possible to classify all C*-algebras without posing any conditions or which classification variants should be adopted for specific classes of C*-algebras. Choose a topic based on your interests. Classification of C*-algebras, for example, is an interesting application of the classification theory to the general theory of C*-algebras.
Use library resources to find published literature on the topic. Elisabeth Blandford, the author of "How to Write the Best Research Paper Ever!," defines research as: "the use of multiple sources and authorities to study, question, and write about a specific topic -- either to discover and publish information, or to persuade an audience to agree with a persuasive thesis concerning that topic." Gather sources, read them, make notes and add them to the Works Cited list, which is the list of sources organized in a specific convention such as the MLA and placed at the end of the paper.
Write a functional introduction laying out the subtopics in a logical order. Write a clear, concise and persuasive thesis sentence -- an educated opinion on the topic indicating your intent to provide evidence in a logical manner. Use a hook such as interesting information to grab the reader's attention. The first systematic classification of approximately finite dimensional C*-algebras, for example, indicates the inductive limits of finite C*-algebras.
Compare, contrast and analyze findings, arranging them in a logical order. Use direct quotes, paraphrases, summaries, examples and illustrations from sources to support your thesis. For example, approximately finite dimensional C*-algebras are completely classified in terms of dimensional groups, i.e., K0-groups. Revise the paper by adding, deleting or changing the order of information. Use words and phrases to promote an even and smooth flow of ideas. Derive the conclusion from the evidence. Classification in terms of K-theoretical invariants, for example, has been derived for C*-algebras.