Explore the foundations of algebraic lattices and Ring Learning with Errors in this comprehensive lecture by Chris Peikert from the University of Michigan, Ann Arbor. Delve into key concepts such as product operations, matrix multiplication, and polynomial representations. Examine the intricacies of dual ideals and their role in lattice-based cryptography. Analyze hardness theorems and their implications for cryptographic security. Investigate ring geometry and its applications in lattice-based systems. Study practical examples that illustrate the theoretical concepts. Uncover the principles of duality in algebraic lattices and their significance. Conclude with an in-depth exploration of hardness proofs, providing a solid understanding of the security foundations in Ring Learning with Errors.